Light-Matter Interactions and Quantum Optics

Home , Density matrix, Dephasing, Dicke model, Field equation, Heisenberg picture, Interaction picture

Light-Matter Interactions and Quantum Optics.

Jonathan Keeling

http://www.tcm.phy.cam.ac.uk/~jmjk2/qo

Contents

Contents iii

Introduction v

1 Quantisation of electromagnetism 1 1.1 Revision: Lagrangian for electromagnetism ...... 2 1.2 Eliminating redundant variables ...... 3 1.3 Canonical quantisation; photon modes ...... 5 1.4 Approximations of light-matter coupling ...... 6 1.5 Further reading ...... 7

2 Quantum electrodynamics in other gauges 9 2.1 Freedom of choice of gauge and classical equations . . . . . 9 2.2 Transformation to the electric dipole gauge ...... 11 2.3 Electric dipole gauge for semiclassical problems ...... 15 2.4 Pitfalls of perturbation ...... 17 2.5 Further reading ...... 18

3 Jaynes Cummings model 19 3.1 Semiclassical limit ...... 19 3.2 Single mode quantum model ...... 20 3.3 Many mode quantum model — irreversible decay ...... 23 3.A Further properties of collapse and revival ...... 27

4 Density matrices for 2 level systems 29 4.1 Density matrix equation for relaxation of two-level system . 29 4.2 Dephasing in addition to relaxation ...... 33 4.3 Power broadening of absorption ...... 36 4.4 Further reading ...... 37

5 Resonance Fluorescence 39 5.1 Spectrum of emission into a reservoir ...... 39 5.2 Quantum regression “theorem” ...... 40 5.3 Resonance ﬂuorescence spectrum ...... 42 5.4 Further reading ...... 44

6 Quantum stochastic methods 47 6.1 Quantum jump formalism ...... 47

iii iv CONTENTS

6.2 Heisenberg-Langevin equations ...... 49 6.3 Fluctuation dissipation theorem ...... 51 6.4 Further reading ...... 52

7 Cavity Quantum Electrodynamics 57 7.1 The Purcell eﬀect in a 1D model cavity ...... 57 7.2 Weak to strong coupling via density matrices ...... 61 7.3 Examples of Cavity QED systems ...... 63 7.4 Further reading ...... 67

8 Superradiance 69 8.1 Simple density matrix equation for collective emission . . . 69 8.2 Beyond the simple model ...... 74 8.3 Further reading ...... 78

9 The Dicke model 79 9.1 Phase transitions, spontaneous superradiance ...... 79 9.2 No-go theorem: no vacuum instability ...... 81 9.3 Radiation in a box; restoring the phase transition ...... 82 9.4 Dynamic superradiance ...... 83 9.5 Further Reading ...... 86

10 Lasers and micromasers 89 10.1 Density matrix equations for a micromaser and a laser . . . 89 10.2 Laser rate equations ...... 92 10.3 Laser Linewidth ...... 93 10.4 Further reading ...... 96

11 More on lasers 99 11.1 Density matrix equation ...... 99 11.2 Spontaneous emission, noise, and β parameter ...... 106 11.3 Single atom lasers ...... 109 11.4 Further reading ...... 112

12 Three levels, and coherent control 113 12.1 Semiclassical introduction ...... 113 12.2 Coherent evolution alone; why does EIT occur ...... 117 12.3 Dark state polaritons ...... 117 12.4 Further reading ...... 120

Bibliography 123 Introduction

The title quantum optics covers a large range of possible courses, and so this introduction intends to explain what this course does and does not aim to provide. Regarding the negatives, there are several things this course deliberately avoids:

• It is not a course on quantum information theory. Some basic notions of coherent states and entanglement will be assumed, but will not be the focus.

• It is not a course on relativistic gauge ﬁeld theories; the majority of solid state physics does not require covariant descriptions, and so it is generally not worth paying the price in complexity of using a manifestly covariant formulation.

• As far as possible, it is not a course on semiclassical electromagnetism. While at times radiation will be treated classically, this will generally be for comparison to a full quantum treatment, or where such an approximation is valid (for at least part of the radiation).

Regarding the positive aims of this course, they are: to discuss how to model the quantum behaviour of coupled light and matter; to introduce some simple models that can be used to describe such systems; to discuss methods for open quantum systems that arise naturally in the context of coupled light and matter; and to discuss some of the more interesting phenomena which may arise for matter coupled to light. Semiclassical behaviour will be discussed in some sections, both because an understanding of semiclassical behaviour (i.e. classical radiation coupled to quantum mechanical matter) is useful to motivate what phenomena might be expected; and also as comparison to the semiclassical case is important to see what new physics arises from quantised radiation. The kind of quantum optical systems discussed will generally consist of one or many few-level atoms coupled to one quantised radiation fields. Realisations of such systems need not involve excitations of real atoms, but can instead be artificial atoms, i.e. well defined quantum systems with discrete level spectra which couple to the electromagnetic field. Such concepts therefore apply to a wide variety of systems, and a variety of characteristic energies of electromagnetic radiation. Systems currently studied experimentally include: real atomic transitions coupled to optical cavities[1];

v vi INTRODUCTION

Josephson junctions in microwave cavities (waveguides terminated by re- flecting boundaries)[2, 3]; Rydberg atoms (atoms with very high principle quantum numbers, hence small differences of energy levels) in GHz cavities[4]; and solid state excitations, 1i.e. excitons or trions localised in quantum dots, coupled to a variety of optical frequency cavities, including simple dielectric contrast cavities, photonic band gap materials, and whispering gallery modes in disks[5]. These different systems provide different opportunities for control and measurement; in some cases one can probe the atomic state, in some cases the radiation state. To describe experimental behaviour, one is in general interested in calculating a response function, relating the expected outcome to the applied input. However, to understand the predicted behaviour, it is often clearer to consider the evolution of quantum mechanical state; thus, both response functions and wavefunctions will be discussed. As such, the lectures will switch between Heisenberg and Schrödingerpic- tures frequently according to which is most appropriate. When considering open quantum systems, a variety of different approaches; density matrix equations, Heisenberg-Langevin equations and their semiclassical approximations, again corresponding to both Schrödinger and Heisenberg pictures. The main part of this course will start with the simplest case of a single two-level atom, and discuss this in the context of one or many quantised radiation modes. The techniques developed in this will then be applied to the problem of many two-level atoms, leading to collective effects. The techniques of open quantum systems will also be applied to describing lasing, focussing on the “more quantum” examples of micromasers and single atom lasers. The end of the course will consider atoms beyond the two-level approximation, illustrating what new physics may arise. Separate to this main discussion, the first two lectures stand alone in discusing where the simple models of coupled light and matter used in the rest of the course come from, in terms of the quantised theory of electromagnetism. Lecture 1

Quantisation of electromagnetism in the Coulomb gauge

Our aim is to write a theory of quantised radiation interacting with quantised matter ﬁelds. Such a theory, e.g. the Jaynes-Cummings model (see next lecture) has an intuitive form:

X † X z + HJ.C. = ωkakak + iσi + gi,kσi ak + H.c. . (1.1) k i,k

† The operator ak creates “a photon” in the mode with wavevector k, and so this Hamiltonian describes a process where a two-level system can change its state with the associated emission or absorption of a photon. The term † ωkakak then gives the total energy associated with occupation of the mode with energy ωk. While the rest of the course is dedicated to studying such models of coupled light-matter system, this (and in part the next) lecture will show the relation between such models and the classical electromagnetism of Maxwell’s equations. To reach this destination, we will follow the route of canonical quantisation; our ﬁrst aim is therefore to write a Lagrangian in terms of only relevant variables. Relevant variables are those where both the variable and its time derivative appear in the Lagrangian; if the time derivative does not appear, then we cannot deﬁne the canonically conjugate momentum, and so cannot enforce canonical commutation relations. The simplest way of writing the Lagrangian for electromagnetism contains irrelevant variables — i.e. the electric scalar potential φ and gauge of the vector potential A; that irrelevant variables exist is due to the gauge invariance of the theory. Since we are not worried about preserving manifest Lorentz covariance, we are free to solve this problem in the simplest way — eliminating the unnecessary variables.

1 2 LECTURE 1. QUANTISATION OF ELECTROMAGNETISM

1.1 Revision: Lagrangian for electromagnetism

To describe matter interacting with radiation, we wish to write a La- grangian whose equations of motion will reproduce Maxwell’s and Lorentz’s equations:

∇ × B = µ0J + µ0ε0E˙ ∇ · E = ρ/ε0 (1.2) ∇ · B = 0 ∇ × E = −B˙ (1.3)

mα¨rα = qα[E(rα) + r˙ α × B(rα)]. (1.4) Equations (1.3) determine the structure of the fields, not their dynamics, and are immediately satisfied by defining B = ∇×A and E = −∇φ−A˙ . Let us suggest the form of Lagrangian L that leads to Eq. (1.2) and Eq. (1.4): Z X 1 ε0 X L = m r˙ 2 + dV E2 − c2B2 + q [r˙ · A(r ) − φ(r )] . 2 α α 2 α α α α α α (1.5) Here, the fields E and B should be regarded as functionals of φ and A. Note also that in order to be able to extract the Lorentz force acting on individual charges, the currents and charge densities have been written as: X X ρ(r) = qαδ(r − rα), J(r) = qαr˙ αδ(r − rα). (1.6) α α The identification of the Lorentz equation is simple: d ∂L d ∂ = [mαr˙ α + qαA(rα)] = mα¨rα + qα(r˙ α · ∇)A(rα) + qα A(rα) dt ∂r˙ α dt ∂t ∂L = = qα∇ [r˙ α · A(rα) − φ(rα)] ∂rα = qα [(r˙ α · ∇) A(rα) + r˙ α × (∇ × A(rα)) − ∇φ(rα)] , thus one recovers the Lorentz equation, ∂ m ¨r = q r˙ × [∇ × A(r )] − ∇φ(r ) − A(r ) . (1.7) α α α α α α ∂t α Similarly, the equation that results from φ can be easily extracted; since ∂L/∂φ˙ = 0, this becomes ∂L X = ε ∇ · E − q δ(r − r ) = 0. (1.8) ∂φ 0 α α α Finally, the equations for A are more complicated, requiring the identity ∂ (∇ × A)2 = 2∇ × (∇ × A), (1.9) ∂A which then gives: d ∂L d ∂L X = − ε E = = −ε c2∇ × (∇ × A) + q r˙ δ(r − r ), dt ˙ dt 0 ∂A 0 α α α ∂A α (1.10) 1.2. ELIMINATING REDUNDANT VARIABLES 3 which recovers the required Maxwell equation

d 1 X − ε E = − ∇ × B + q r˙ δ(r − r ). (1.11) 0 dt µ α α α 0 α

Thus, the Lagrangian in Eq. (1.5), along with the deﬁnitions of E and B in terms of A and φ produce the required equations.

1.2 Eliminating redundant variables

As mentioned in the introduction, we must remove any variable whose time derivative does not appear in the Lagrangian, as one cannot write the required canonical commutation relations for such a variable. It is clear from Eq. (1.5) that the electric scalar potential φ is such a variable. Since φ˙ does not appear, it is also possible to eliminate φ directly from the equation ∂L/∂φ; using Eq. (1.8) and the definition of E, this equation gives: 2 − ε0∇ · A˙ − ε0∇ φ − ρ(r) = 0. (1.12) Rewriting this in Fourier space, one has: 1 ρ(k) ˙ φ(k) = 2 + ik · A(k) . (1.13) k ε0 We can now try to insert this definition into the Lagrangian, to eliminate φ. To do this, we wish to write E2 and B2 in terms of φ and A; it is therefore useful to start by writing ˙ kj kjkk ˙ − Ej(k) = ikjφ(k) + Aj(k) = i 2 ρ(k) + δjk − 2 Ak(k). (1.14) ε0k k This means that the electric field depends on the charge density, and on the transverse part of the vector potential, which will be written:

k k A⊥k = δ − j k A˙ (k). (1.15) j jk k2 k

The transverse1 part of the vector potential is by deﬁnition orthogonal to the wavevector k, and so the electric ﬁeld is the sum of two orthogonal vectors, and so: 2 ˙ ⊥ 2 1 2 |E(k)| = |A (k)| + 2 2 |ρ(k)| . (1.18) ε0k

1The combination: „ k k « δ⊥ (k) = δ − j k , (1.16) jk jk k2 is the reciprocal space representation of the transverse delta function; with appropriate regularisation[6, Complement AI ], it can be written in real space as: 2 1 “ r r ” δ⊥ (r) = δ(r)δ + 3 j k − δ . (1.17) jk 3 jk 4πr3 r2 jk 4 LECTURE 1. QUANTISATION OF ELECTROMAGNETISM

Similarly, the squared magnetic ﬁeld in reciprocal space is given by: |B(k)|2 = (k × A(k)) · (k × A∗(k)) k k = k2 δ − j k A (k)A∗(k) = k2|A⊥(k)|2. (1.19) jk k2 j k Thus, the ﬁeld part of the Lagrangian becomes: Z Z ε0 2 2 2 1 3 1 2 dV E − c B = − d k 2 |ρ(k)| 2 ε0 k Z 3 ⊥ 2 2 2 ⊥ 2 + ε0− d k |A˙ (k)| − c k |A (k)| . (1.20)

The notation R− has been introduced to mean integration over reciprocal half-space; since A(r) is real, A∗(k) = A(−k), thus the two half spaces are equivalent. This rewriting is important to avoid introducing redundant fields in the Lagrangian; the field is either specified by one real variable at all points in real space, or by two real variables at all points in reciprocal half-space. Similar substitution into the coupling between fields and matter, written in momentum space gives: Z 3 ∗ ∗ Lem−matter = 2<− d k [J(k) · A (k) − ρ(k)φ (k)] Z ∗ 3 ∗ ρ (k) ik ˙ ∗ = 2<− d k J(k) · A (k) − ρ(k) 2 − 2 · A (k) . ε0k k (1.21) This can be simplified by adding a total time derivative, L → L + dF/dt; such transformations do not affect the equations of motion, since they add only boundary terms to the action. If: Z −ik · A∗(k) F = 2<− d3kρ(k) , (1.22) k2 then Z 2 dF 3 ik ∗ |ρ(k)| Lem−matter + = 2<− d k J(k) − ρ˙(k) 2 · A (k) − 2 . dt k ε0k (1.23) Then, using conservation of current,ρ ˙(k) + ik · J(k) = 0, one finally has: Z 2 dF 3 ⊥ ⊥∗ |ρ(k)| Lem−matter + = 2<− d k J (k) · A (k) − 2 . (1.24) dt ε0k Note that this set of manipulations, adding dF/dt has eliminated the longitudinal part of the vector potential from the Lagrangian. The form chosen for F is such that this procedure is equivalent to a gauge transformation; the chosen gauge is the Coulomb gauge. Putting everything together, one has: X 1 1 Z 1 L = m r˙ 2 − − d3k |ρ(k)|2 coulomb 2 α α ε k2 α 0 Z 3 ⊥ 2 2 2 ⊥ 2 h ⊥ ⊥∗ i + ε0− d k |A˙ (k)| − c k |A (k)| + 2< J (k) · A (k) . (1.25) 1.3. CANONICAL QUANTISATION; PHOTON MODES 5

Thus, the ﬁnal form has divided the interaction into a part mediated by transverse ﬁelds, described by A⊥, and a static (and non-retarded) Coulomb interaction. Importantly, there are no irrelevant variables left in Eq. (1.25) The Coulomb term can also be rewritten: Z 1 3 1 2 X qαqβ Vcoul = − d k 2 |ρ(k)| = . (1.26) ε0 k 8πε0|rα − rβ| α,β Note that since the Coulomb interaction is non-retarded, both the Coulomb and transverse parts of interaction must be included to have retarded interactions between separated charges.

1.3 Canonical quantisation; photon modes

We have in Eq. (1.25) a Lagrangian which can now be treated via canonical quantisation. Since only the transverse part of the ﬁeld A appears in Eq. (1.25), we can drop the superscript label in A⊥ from here on. To proceed, we should ﬁrst identify the canonical momenta and the Hamiltonian, and then impose canonical commutation relations. Thus,

∂Lcoulomb pα = = mr˙ α + qαA(rα) (1.27) ∂r˙ α ∂Lcoulomb Π(k) = = ε0A˙ (k). (1.28) ∂A˙ (k)∗ P ˙ Then, constructing the Hamiltonian by H = i PiRi − L, one ﬁnds: X 1 H = [p − q A(r )]2 + V 2m α α α coul α α Z 2 3 |Π(k)| 2 2 2 + ε0− d k 2 + c k |A(k)| . (1.29) ε0 In order to quantise, it remains only to introduce commutation relations for the canonically conjugate operators. Noting that A(r) has only two independent components, because it is transverse, it is easiest to write its commutation relations in reciprocal space, introducing directions ek,n orthogonal to k with n = 1, 2 ; then:

[ri,α, pj,β] = i~δαβδij (1.30) h 0 i 0 0 Aek,n (k), Πek0,n0 (k ) = i~δ(k − k )δnn . (1.31) This concludes the quantisation of matter with electromagnetic interactions in the Coulomb gauge. It is however instructive to rewrite the transverse part of the ﬁelds in terms of their normal modes. The second line of Eq. (1.29) has a clear similarity to the harmonic oscillator, with a separate oscillator for each polarisation and momentum. Rewriting in normal modes thus means introducing the ladder operators: r ε0 i ak,n = ckAek,n (k) + Πek,n (k) , (1.32) 2~ck ε0 6 LECTURE 1. QUANTISATION OF ELECTROMAGNETISM or, inverted one has: s X X ~ ik·r † −ik·r A(r) = eˆn ak,ne + ak,ne . (1.33) 2ε0ωkV k n=1,2

Inserting this into Eq. (1.29) gives the ﬁnal form:

X 1 2 X X † H = [pα − qαA(rα)] + ~ωkakak + Vcoul. (1.34) 2mα α k n=1,2

1.4 Dipole, two-level, and rotating wave approximations

Equation (1.34) applies to any set of point charges interacting with the electromagnetic field. In many common cases, one is interested in dipoles, with pairs of opposite charges closely spaced, and much larger distances between the dipoles. In this case, there are a number of approximations one can make to simplify calculations. This section will briefly illustrate these approximations: neglect of the A2 terms, the dipole approximation, projection to two-level systems, and the rotating wave approximation. The study of the approximate models that result will be the subject of the rest of this course. 2 2 Neglect of the A terms in expanding [pα − qαA(rα)] can be justified in the limit of low density of dipoles; considering only a single radiation mode, the contribution of the A2 term can be rewritten using Eq. (1.33) as: 2 2 N q ~ † δHA2 = a + a . (1.35) V 4mε0ω Thus, this term adds a self energy to the photon field, which scales like the density of dipoles. The relative importance of this term can be estimated by comparing it to the other term in the Hamiltonian which is quadratic † in the photon operators, ~ωkakak. Their ratio is given by:

2 2 N q N 3 Ryd 2 ∝ aB (1.36) V 4mε0(ωk) V ~ωk thus, assuming particles are more dilute than their Bohr radius, neglect of δHA2 is valid for frequencies of the order of the Rydberg for the given bound system of charges. Turning next to the dipole approximation, consider a system with just two charges: charge +q mass m1 at R + r/2 and −q mass m2 at R − r/2. If r λ where λ is a characteristic wavelength of light, then one may assume that A(R + r/2) ' A(R − r/2) ' A(R), and so the remaining coupling between radiation and matter is of the form p1 p2 δHA·p = q − · A(R). m1 m2 1.5. FURTHER READING 7

Then, in the case that one can write H0, a Hamiltonian for the dipole without its coupling to radiation one can use p/m =x ˙ = i[H0, x]/~, thus giving:

q X † H = H0 − i [H0, r] · A(R) + ~ωkakak + Vcoul. (1.37) ~ k,n=1,2 To further simplify, one can now reduce the number of states of the dipoles that are considered; currently, there will be a spectrum of eigenstates of H0, and transitions are induced between these states according to hψf |r|ψii. Finally, restricting to only the two lowest atomic levels and to a single radiation mode, one has a model of two-level systems (describing matter) coupled to bosonic modes (describing radiation). This model is known as the Jaynes-Cummings model:

1 g(a + a†) H = + ω a†a. (1.38) 2 g(a + a†) − ~ 0

Here we have introduced the energy splitting between the lowest two atomic levels. In terms of the upper and lower states, |ai and |bi, the atom-photon coupling strength g can be written:

gk 1 ek,n · dba = q hb|[H0, r]|ai √ = √ , (1.39) 2 2ε0~ωkV 2ε0~ωkV where dba = q hb|r|ai is the dipole matrix element. The matrix notation represents the two levels of the dipole. We assumed H0 commutes with the parity operator x → −x, and so the coupling to radiation appears only in the off-diagonal terms in the two-level basis. The final approximation to be discussed here, the rotating wave approximation, is appropriate when ' ω0. Considering g as a perturbation, one can identify two terms: g 0 a g 0 a† ∆ = , ∆ = , (1.40) co 2 a† 0 cross 2 a 0 where ∆co, the co-rotating terms, “conserve energy”; and ∆cross do not. More formally, the effects of ∆cross give second order perturbation terms 2 2 like g /(ω0 + ), while ∆co give the much larger g /(ω0 − ).

1.5 Further reading

The discussion of quantisation in the Coulomb gauge in this chapter draws heavily on the book by Cohen-Tannoudji et al. [6].

Questions

Question 1.1: Transverse delta function Prove that Eq. (1.17) is the Fourier transform of Eq. (1.16). It is helpful to consider a modiﬁed version of Eq. (1.17), with a factor exp(−mk2), and take m → 0 only at the end of the calculation. 8 LECTURE 1. QUANTISATION OF ELECTROMAGNETISM

Question 1.2: Thomas-Reiche-Kuhn sum rule Prove the equality:

2δ [r , [H , r ]] = ~ αβ , (1.41) α 0 β m stating the most general form of H0 for which this is true. From the expectation of this commutator, show that the dipole matrix elements between any state a and all other states b obeys the relation:

X X 2q2 |d |2(E − E ) = ~ α (1.42) ab b a 2m b α . Lecture 2

Gauge transformations, and quantum electrodynamics in other gauges

In the previous section, we choose to work in the Coulomb gauge, adding a total time-derivative to the Lagrangian which had the eﬀect of removing all dependence of the Lagrangian on Ak. This choice, which is equivalent to choosing to impose Ak ≡ 0 hugely simpliﬁed the subsequent algebra, but is not strictly necessary. This chapter describes the consequences of making other choices for Ak; such choices turn out to correspond to making a unitary transformation in the quantum problem. In the special case of one or a few localised systems of charge — where charges within a system are separated by far less than the wavelength of light — a change of gauge to the electric dipole gauge can simplify calculations, and provides further understanding of the relation between the instantaneous Coulomb interaction and the photon-mediated terms in Eq. (1.34).

2.1 Freedom of choice of gauge and classical equations

The classical Lagrangian in the previous section depended on the longitudinal part of the vector potential only in the coupling between matter and electromagnetic ﬁelds, Eq. (1.21)

Z 3 ∗ ∗ Lem-matter = 2<− d k [J(k) · A (k) − ρ(k)φ (k)] Z ∗ 3 ∗ ρ (k) ik ˙ ∗ = 2<− d k J(k) · A (k) − ρ(k) 2 − 2 · A (k) . ε0k k (1.21)

9 10 LECTURE 2. QED IN OTHER GAUGES

Using the continuity equationρ ˙ + ik · J = 0 to eliminate Jk = iρ/k˙ and breaking A into transverse and longitudinal parts yields Z 2 3 ∗ i ∗ |ρ(k)| i ˙ ∗ Le-m = 2<− d k J⊥(k) · A⊥(k) + ρ˙(k)Ak(k) − 2 + ρ(k)Ak(k) . k ε0k k (2.1) It is therefore clear that taking any functional form for Ak = g(A⊥) leads to a change to the Lagrangian which may be written as: d Z i δL = 2<− d3k ρg∗ (A ) . (2.2) gauge dt k ⊥

The choice of Ak is the choice of gauge; by choosing Ak as a function of A⊥, one can ensure, for example, that n · A(r) = 0 for some fixed n. Since such a condition corresponds only to an additional total derivative, the action changes only by boundary terms, and so the equations of motion are not affected. The definition of canonical momentum however does change. Rewriting Z Z ∗ ∂ 3 i ∗ 3 i ∂g (A⊥) ˙ ∗ δLgauge = 2<− d k ρg (A⊥) + 2<− d k ρ ∗ A⊥(k) , ∂t k k ∂A⊥ (2.3) the canonical momentum thus becomes: ∂L iρ(k) ∂g∗ (A ) Π(k) = = Π (k) + ⊥ (2.4) ˙ ∗ Coulomb ∂A⊥(k) k ∂A⊥

Quantum formalisms resulting from different gauges Once we have found the canonical momentum in a given gauge, we can quantise in this gauge by promoting the dynamical variables and their canonical conjugates to operators, and imposing canonical commutation re- 0 0 lations. This means [A⊥n(k), Πn0 (k )] = i~δ(k − k )δnn0 , with Π(k) being the new canonical momentum in the new gauge: The canonical momentum corresponds to a different combination of physical fields in different gauges. If we write Z i F (A , ρ) = 2<− d3k ρg∗ (A ) , (2.5) ⊥ k ⊥ then we may compare the two formalisms:

A⊥,new(k) =A⊥,old(k) (2.6) ∂F Πnew(k) =Πold(k) + . (2.7) ∂A⊥,new(k) If we consider a state |πi which is an eigenstate of the old momentum, Πold(k)|πi = λπ|πi, then in the new formalism, this state obeys: ∂F λπ|πi = Πnew(k) − |πi (2.8) ∂A⊥,new(k) ∂ ∂F = −i~ − |πi (2.9) ∂A⊥,new(k) ∂A⊥,new(k) iF (A ,ρ)/ ∂ −iF (A ,ρ)/ = e ⊥ ~ −i~ e ⊥ ~|πi. (2.10) ∂A⊥,new(k) 2.2. TRANSFORMATION TO THE ELECTRIC DIPOLE GAUGE 11

Hence, the eigenstates of the momenta of the old and new formalism are related by the unitary transform exp[−iF (A⊥, ρ)/~] — one can immediately see that this relation of eigenstates of operators works for all canonical momenta, as well as working for the position operators which commute with the unitary transforms. It is worth stressing that the statement proved here is that all gauge transforms correspond to unitary transforms of the quantum formalism, and not the other way around. It is also worth highlighting (as was implicit in the calculation), that this occurred because the gauge transforms corresponded classically to canonical transformations of the La- grangian (addition of a total time derivative), and that it is more generally true that canonical transformations of classical dynamical systems correspond to unitary transformations of the associated quantum problem.

Formal equivalence of gauges Formally, since a gauge transformation corresponds to a unitary transformation, calculations of physical quantities in different gauges should exactly match, i.e. D E D E D E (1) (1) (1) (1) † (1) † (1) (2) (2) (2) ψ O ψ = ψ U UO U U ψ = ψ O ψ . (2.11) Despite this formal equivalence, there have been long and convoluted arguments about gauge invariance in approximate methods. One particularly dangerous feature of such transformations is that the “natural” identification of a bare and perturbation Hamiltonian can be different between different gauges (this is true of the discussion below), and so eigenstates of the bare Hamiltonian are not a gauge invariant quantity. Two general solutions to such problems are

• To use a basis of states that correspond to eigenstates of a physical (i.e. gauge invariant) operator, such as the mechanical momentum, physical ﬁelds, and the total energy.

• To use only the formal equivalence, and demand a transformation of states as well as of operators when switching between gauges.

2.2 Transformation to the electric dipole gauge

The preceeding working allows us to directly transform between two quantum pictures, rather than needing to changes gauges classically and re- quantize. As an illustration of this in a case where it is particularly clear, this section describes the transformation to the electric dipole gauge for a problem of two systems of localised charges, where each system is overall neutral. We consider a hierarchy of scales, such that within a given system all charges are within a distance l and l λ, |RA − RB|, where RA,B are the centres of mass of the two systems. This is illustrated in Fig. 2.1. Since l λ, we can approximate the electromagnetic ﬁelds as being those at the centre of mass of each system. In the Coulomb gauge, this 12 LECTURE 2. QED IN OTHER GAUGES λ RA − RB l Figure 2.1: Two localised systems, consisting of charges which are closely spaced in each system. then can be written following the notation in Eq. (1.34) as:

X 1 2 X 1 2 H = [pα − qαA⊥(RA)] + [pβ − qβA⊥(RB)] 2mα 2mβ α β X † AA BB AB + ωk~akak + Vcoul + Vcoul + Vcoul, (2.12) k,n in which the Coulomb interaction has been divided into parts limited to each subsystem as well as a cross term. This cross term is instantaneous (as all the Coulomb terms are), while the real physical interaction is retarded. To recover this retardation it is however necessary to include the transverse parts of the ﬁelds. From this point onwards we will set ~ = 1

Transformation of Hamiltonian The unitary transformation that will simplify the results is of the form:

X U = exp [idA · A⊥(RA) + idB · A⊥(RB)] , dA = qα(rα − RA), α (2.13) in which we have introduced the dipole moments dA,B of the two systems. This transformation clearly commutes with the position operator, and ought to commute with the transverse vector potential (this is shown to be true explicitly below). However, it does not commute with pα nor † ak, ak. One instead ﬁnds: ! ∂ X U †p U = p − i i q (r − R ) · A (R ) α α ∂r α α A ⊥ A α α ! ∂RA X l = pα + qαA⊥(RA) − qα0 · A⊥(RA) + O . ∂rα λ α0 (2.14)

P In this, the term α qα = 0 due to neutrality of each subsystem, and the higher order terms in l/λ are ignored due to the long wavelength approxi- † mation. Hence all that remains is U pαU ≈ pα + qαA⊥(RA). 2.2. TRANSFORMATION TO THE ELECTRIC DIPOLE GAUGE 13

For the photon annihilation operators no approximations are needed. The unitary transformation can be rewritten as    X eˆn h i  U = exp i √ · d e−ik·RA + d e−ik·RB a† + H.c. , (2.15) 2ε ω V A B k  k,n 0 k  which has the form of a shift operator for the annihilation operators:

eˆ h i † n −ik·RA −ik·RB U akU = ak + i√ · dAe + dBe (2.16) 2ε0ωkV

From this we can reconﬁrm that the unitary matrix commutes with the transverse vector potential,

X eˆn h i A (r) = √ e−ik·ra† + eik·ra , (2.17) ⊥ 2ε ω V k k k,n 0 k hence,

† U A⊥(r)U = A⊥(r) ieˆ h i X n ik·(r−RA) ik·(r−RB ) + eˆn · dAe + eˆn · dBe − H.c. . (2.18) 2ε0ωkV k,n

It is clear that due to the antisymmetry of the sum under k ↔ −k the sum vanishes, so A⊥(r) commutes with the transformation. Hence, the transformed form of the Hamiltonian is given by:

2 p2 ˜ † X pα X β AA BB AB H = U HU = + + Vcoul + Vcoul + Vcoul 2mα 2mβ α β     eˆ · d X † X n S −ik·RS † ik·RS + ωk a ak + i√ e a − e a  k 2ε ω V k k k,n  S=A,B 0 k ) 1 h i 2 2 ik·(RA−RB ) + (eˆn · dA) + (eˆn · dB) + (eˆn · dA)(eˆn · dB) 2e , 2ε0V (2.19) where the last line has relabelled the sum via k ↔ −k.

Transformation of physical ﬁelds To simplify this expression, we need two connected things:

• To identify the combination of photon operators that appears in the second line as the light-matter coupling.

• To simplify the cross terms between the A and B systems. 14 LECTURE 2. QED IN OTHER GAUGES

It is tempting to use the definition of the transverse electric field in the old gauge, r X ωk h −ik·r † ik·r i E⊥(r) = i eˆn e ak − e ak , (2.20) 2ε0V k,n to identify the combination of operators, however this relation between the physical operators and the annihilation operators is true only in the Coulomb gauge, not the electric dipole gauge. Transforming this electric field using Eq. (2.16) gives:

eˆ h h i i † X n ik·(r−RA) ik·(r−RB ) U E⊥U = E⊥ − eˆn · dAe + dBe + H.c. . 2ε0V k,n (2.21) This sum is symmetric under k ↔ −k and so does not vanish. Instead it can be identified in terms of the polarisation. Since the dipole matrix element of system A is dA, we can write PA(r) = dAδ(r − RA); this places the polarisation at the centre of charges, and is measured w.r.t a reference state in which all the charges are at the centre of mass, i.e. ∀rα = RA. If we then use the transverse delta function to find the transverse part of this expression, we find that: 1 X h i P (r) = eˆ eˆ · d eik·(r−RA) (2.22) A,⊥ V n n A k,n and hence: r X ωk h −ik·r † ik·r i P⊥(r) D⊥(r) i eˆn e ak − e ak = E⊥(r) + = (2.23) 2ε0V ε0 ε0 k,n This solves our first question; the second is also considerably simplified when we note that the term: 1 X ik·(RA−RB ) (eˆn · dA)(eˆn · dB) e = ε0V k,n     Z d3r 1 h i 1 h i X ik·(r−RA) X ik·(r−RB )  eˆn eˆn · dAe  ·  eˆn eˆn · dBe  ε0 V V k,n k,n Z d3r = PA,⊥(r)PB,⊥(r). (2.24) 0 Thus, we can write:

2 p2 ˜ X pα X β AA BB AB H = + + Vcoul + Vcoul + Vcoul 2mα 2mβ α β X † + ωkakak + D⊥(RA) · dA + D⊥(RB) · dB k,n Z 3 A B d r + Σdipolar + Σdipolar + PA,⊥(r)PB,⊥(r). (2.25) 0 2.3. ELECTRIC DIPOLE GAUGE FOR SEMICLASSICAL PROBLEMS 15 Instantaneous nature of interaction To complete the simpliﬁcation, it is necessary to write the cross-term in the Coulomb interaction similarly to Eq. (2.24). The Coulomb interaction arose by writing the longitudinal part of the electric ﬁeld in terms of the charge density. Inverting this procedure easily gives:

ε Z h i V AB = 0 d3r E (r) + E (r)2 − E (r)2 − E (r)2 (2.26) coul 2 k,A k,A k,A k,A Z Z 3 1 3 = ε0 d rEk,A(r) · Ek,A(r) = d rPk,A(r) · Pk,A(r). (2.27) ε0

The last step made use of the reference distribution of charges; since this distribution has no net charges (each system is neutral, and all charges coincide), then ∇ · D = 0, i.e. Dk = 0, so Ek = Pk/ε0. This allows us to combine:

Z 3 Z 3 AB d r d r Vcoul + PA,⊥(r)PB,⊥(r) = PA(r)PB(r) = 0 (2.28) 0 0 where the last line follows from the deﬁnition of the polarisation, which is zero outside each region of charge. Hence, the ﬁnal Hamiltonian is:

2 p2 ˜ X pα X β AA BB A B H = + + Vcoul + Vcoul + Σdipolar + Σdipolar 2mα 2mβ α β X † + ωkakak + D⊥(RA) · dA + D⊥(RB) · dB. (2.29) k,n

This transformation has therefore removed the instantaneous interaction between the two systems; all interaction is now described by the quantised D field. Since Dk = 0, the transverse and total fields are equivalent, hence the transverse D field is physical and retarded.

2.3 Electric dipole gauge for semiclassical problems

The previous section showed the unitary transformation to the electric dipole gauge for quantised radiation coupled to quantised matter. For the simpler case of a single localised system, the transformation would be U = exp[id · A⊥(0)], yielding:

2 ˜ X pα X † H = + Vcoul + Σdipolar + ωkakak + D⊥(0) · d. (2.30) 2mα α k,n

For comparison, this section describes the even simpler transformation that applies when the radiation is assumed to be classical; i.e. it has no 16 LECTURE 2. QED IN OTHER GAUGES

free dynamics, but rather the operator A⊥ is replaced by a time dependent ﬁeld corresponding to some incident radiation. In this case the initial Hamiltonian is:

X 1 H = [p − q A (0, t)]2 + V . (2.31) s.c. 2m α α ⊥ coul α α

The gauge transformation is formally the same as before, but as now A⊥ is not an operator but a time-dependent ﬁeld we must use the result:

The transformation of pα is exact as in Eq. (2.14). Since A⊥ is a c- number, it commutes with the unitary transformation, and so the ﬁnal result is:

X p2 H˜ = α + V − i∂ [id · A (0)] s.c. 2m coul t ⊥ α α X p2 = α + V − d · E (0) (2.35) 2m coul ⊥ α α

Relating S-matrix elements in diﬀerent gauges In the semiclassical case, as well as the formal equivalence between gauges, there are “less formal” relations that can be shown to hold. One example arises if we consider a ﬁeld A⊥(t) = A0λ(t) cos(ωt), where λ(t) is a function that goes to 0 at ±∞, but λ(t) = 1 for t1 < t < t2, with smooth interpola- tion between: Since this means U(t) = exp[iλ(t)d · A0 cos(ωt)], it is clear λ(t)

Figure 2.2: Form of λ(t) for S-matrix that at long times U(t) = 1, and hence eigenstates in the two gauges are coincident. This allows us to consider the S-matrix:

iH0t2 −iH0t1 Sab = hb| e U(t2, t1)e |ai . (2.36)

Since U = 1 at long times the bare Hamiltonian H0 is also the same between the two gauges, so this S-matrix should be gauge invariant. Let us consider its calculation perturbatively in the two gauges 2.4. PITFALLS OF PERTURBATION 17

Calculation in A · p gauge

In the Coulomb gauge, the perturbation Hamiltonian (to ﬁrst order in A0) is given by: X pαqα H (t) = − · A cos(ωt). (2.37) int m 0 α α Using the result of time-dependent perturbation theory:

Z t2 −iH0(t2−τ) −iH0(τ−t1) U(t2, t1) = 1 − i dτe Hint(τ)e + ... (2.38) t1 we have:

= δab − π [δ(ωba + ω) + δ(ωba − ω)] ωbaA0 · hb| d |ai . (2.40)

In the last line we have used the fact that the bare Hamiltonian depends 2 only on pα via pα/2mα, allowing one to rewrite the matrix element of momentum in terms of the dipole matrix element.

Calculation in E · r gauge In the electric dipole gauge, the perturbation is instead

Hint(t) = −d · E(t). (2.41) where E(t) = −A˙ (t) ≈ A0ω sin(ωt), and so starting from Eq. (2.39) we have:

Z t2 iωbaτ Sab = δab + i dτe sin(ωt)ωA0 · hb| d |ai t1 = δab − π [δ(ωba − ω) − δ(ωba + ω)] ωA0 · hb| d |ai , (2.42) where we have made the same resonant approximation as above. Although functionally diﬀerent, the two expressions do match, because ωba = ω is ensured by the delta function. For a further example of this kind of equivalence, see question 2.1.

2.4 Pitfalls of perturbation

The idea of writing the free Hamiltonian H0 is in fact rather subtle to deﬁne correctly: Atoms are held together by electromagnetic forces, and as mentioned above, separation into transverse, photon mediated, and Coulomb 18 LECTURE 2. QED IN OTHER GAUGES

terms leads to apparently non-retarded interactions. Further, H0 diﬀers between diﬀerent gauges. The question of most practical relevance is how the parameters in a semi-phenomenological model of a coupled light-matter system should be related to experimentally measured quantities.

2.5 Further reading

The majority of this chapter is also based on discussions in the book by Cohen-Tannoudji et al. [6]. The subtlety mentioned briefly, about how one can define a “free Hamiltonian” for atoms when the binding within an atom is due to the electromagnetic field has been discussed in the context of which gauge should be used in calculating off-resonant transitions. See for example [7–9].

Questions

Question 2.1: Two-photon transitions If we choose states |bi, |ai such that hb| d |ai = 0, then the leading order contribution to the transition comes from two photon processes. Assuming a 6= b, the leading order contribution to the S-matrix can be written:

t2 τ Z Z 0 0 iωbcτ iωcaτ 0 Sab = − dτ dτ e e hb| Hint(τ) |ci hc| Hint(τ ) |ai (2.43) t1 t1

2.1.(a) Making the resonant approximation, show that the expressions for Sab in the two gauges can be written as:

δ(ωba − 2ω) X ωbcωca SA·p = 2π A · hb| d |ci A · hc| d |ai (2.44) ab 4 ω − ω 0 0 c ca 2 δ(ωba − 2ω) X ω SE·r = 2π A · hb| d |ci A · hc| d |ai (2.45) ab 4 ω − ω 0 0 c ca

2.1.(b) Prove the equivalence of these expressions by showing that:

2 X ωbcωca − ω D = · hb| d |ci · hc| d |ai = 0 (2.46) ω − ω i j c ca if 2ω = ωba. [Hint: use the two-photon resonance condition to show that X D ∝ (ωbc − ωca) · hb| di |ci · hc| dj |ai , (2.47) c then rewrite this sum in terms of the commutator [[H0, di], dj].] Lecture 3

One two-level system coupled to photons: the Jaynes-Cummings model

The aim of this first section is to discuss the interaction between a single two-level system and various different configurations of photon fields. We will begin by discussing the interaction with a semiclassical light field, then consider a single quantised mode of light in an initially coherent state, and finally consider a continuum of quantised modes. In future lectures we will then consider combinations of these cases, to describe realistic cavity QED systems, in which the cavity picks out a particular mode, but coupling to a continuum of background modes also exists. Even in the case of a classical applied field, the behaviour of a two-level system is not entirely trivial, as the two-level system is non-linear, i.e. it does not have an harmonic spectrum.

3.1 Semiclassical limit

On applying a classical light field to a two-level system, we can adapt the Jaynes-Cummings model by neglecting the dynamics of the photon field, and replacing the photon creation and annihilation operators with the amplitude of the time-dependent classical field. This gives the effective Hamiltonian: 1 gλe−iω0t H = . (3.1) 2 gλeiω0t −

We have made the rotating wave approximation, and hence only included the positive/negative frequency components of the classical ﬁeld in the rais- ing/lowering operators for the two-level system. To solve this problem, and ﬁnd the time-dependent state of the two-level atom, it is convenient to make a transformation to a rotating frame. In general, if

eif(t) 0 U = , (3.2) 0 e−if(t)

19 20 LECTURE 3. JAYNES CUMMINGS MODEL then ˙ −2if(t) ˜ H11 + f(t) H12e H → H = 2if(t) , (3.3) H21e H22 − f˙(t) thus, in this case we use f = −ωt/2, giving:

1 − ω gλ H = . (3.4) 2 gλ −( − ω)

Since this transformation affects only the phases of the wavefunction, we can then find the absorption probability by considering the time averaged probability of being in the upper state. The eigenvalues of Eq. (3.2) are ±Ω/2 where Ω = p( − ω)2 + g2λ2. Writing the eigenstates as (cos θ, sin θ)T , and (− sin θ, cos θ)T corresponding respectively to the ± roots (assuming tan(θ) > 0). Substituting these forms, one finds the condition: tan(2θ) = gλ/( − ω). This form is sufficient to find the probability of excitation. If the two-level system begins in its ground state, then the time-dependent state can be written as:

ψ cos θ − sin θ ↑ = sin θ e−iΩt/2 + cos θ eiΩt/2 (3.5) ψ↓ sin θ cos θ

1 − cos(Ωt) (gλ)2 P = |ψ |2 = |2 sin θ cos θ sin(Ωt/2)|2 = . ex ↑ 2 (gλ)2 + ( − ω)2 (3.6) Thus, the probability of excitation oscillates at the Rabi frequency Ω = p 2 2 ( − ω) + (gλ) , and the amplitude of oscillation depends on detuning.√ On resonance ( = ω), one can engineer a state with |ψ↑| = |ψ↓| = 1/ 2, or a state with |ψ↑| = 1 by applying a pulse with a duration (gλ)t = π/2 or π respectively.

3.2 Single mode quantum model

Let us now repeat this analysis in the case of a quantised mode of radiation, starting with the Jaynes-Cummings model in the rotating wave approximation. 1 ga H = + ω a†a. (3.7) 2 ga† − 0

Rabi oscillations in the Jaynes-Cummings model

In the rotating wave approximation the total number of excitations Nex = σz + a†a is a conserved quantity, thus if we start in a number state of the light ﬁeld: (a†)n |n, ↓i = √ |0, ↓i , (3.8) n! then the general state at later times can always be written:

|ψ(t)i = α(t) |n − 1, ↑i + β(t) |n, ↓i . (3.9) 3.2. SINGLE MODE QUANTUM MODEL 21

Inserting this ansatz in the equation of motion gives √ α(t) 1 1 ( − ω) g n α(t) i∂ = n − ω1 + √ . t β(t) 2 2 g n −( − ω) β(t) (3.10) After removing the common phase variation exp[−i(n−1/2)ωt[, the general solution can be written in terms of the eigenvectors and values of the matrix in Eq. (3.10), and reusing the results for the semiclassical case, we have the state α(t) cos θ − sin θ = e−i(n−1/2)ωt sin θe−iΩt/2 + cos θeiΩt/2 , β(t) sin θ cos θ (3.11) and the excitation probability is thus: 1 − cos(Ωt) g2n P = (3.12) ex 2 g2n + ( − ω)2 √ where we now have tan(2θ) = g n/( − ω) and Ω = p( − ω)2 + g2n.

Collapse and revival of Rabi oscillations Restricting to the resonant case = ω, let us discuss what happens if the initial state does not have a well defined excitation number. Since the oscillation frequency depends on n, the various components of the initial state with different numbers of excitations will oscillate at different frequencies. We can therefore expect interference, washing out the signal. This is indeed seen, but in addition one sees revivals; the signal reappears at much later times, when the phase difference between the contributions of succesive number state components is an integer multiple of 2π. Let us consider explicitly how the probability of being in the excited state evolves for an initially coherent state n 2 2 X λ |ψi = e−|λ| /2 exp(λa†) |0, ↓i = e−|λ| /2 √ |n, ↓i . (3.13) n n! If resonant, the general result in Eq. (3.11) simplifies, as θ = π/4. Following the previous analysis, the state at any subsequent time is given by:

2 |ψi = e(−|λ| +iωt)/2 √ √ X (λe−iωt)n g nt g nt × √ cos |n, ↓i + sin |n − 1, ↑i . (3.14) 2 2 n n! Thus, the probability of being in the excited atomic state, traced over all possible photon states, is given by 2n √ X 2 X |λ| g nt P = |hn, ↑ |ψi|2 = e−|λ| sin2 ex n! 2 n n 2n 1 1 2 X |λ| √ = − e−|λ| cos g nt . (3.15) 2 2 n! n 22 LECTURE 3. JAYNES CUMMINGS MODEL

We consider the case λ 1. In this case, one may see that the coefficients |λ|2n/n! are sharply peaked around n ' |λ|2 by writing: |λ|2n 1 = √ exp[−f(n)], f(n) = n ln(n) − n − n ln(|λ|2), (3.16) n! 2πn By differentiating f twice, one finds f 0(n) = ln n/|λ|2, f 00 = 1/n, and hence one may expand f(n) near its minimum at n = |λ|2 to give: x2 f(n = |λ|2 + x) ' −|λ|2 + (3.17) 2|λ|2 Using this in Eq. (3.15) gives the approximate result:

( 2 ) 1 1 X m p 2 Pex ' − < exp − + igt |λ| + m , (3.18) 2 p 2 2|λ|2 2 2π|λ| m where it is assumed that |λ|2 is large enough that the limits of the sum may be taken to ±∞. Three different timescales affect the behaviour of this sum. Concentrat- ing on the peak, near m = 0, there is a fast oscillation frequency gλ, describing Rabi oscillations. Then, considering whether the terms in the sum add in phase or out of phase, there are two time scales: collapse of oscillations occurs when there is a phase difference of 2π across the range of terms p 2 |m| < σm contributing to this sum, i.e. when gTcollapse( |λ| + σm −|λ|) = π. Taking σm = |λ| from the Gaussian factor, this condition becomes gTcollapse ' 2π. On the other hand, if the phase difference between each successive term in the sum is 2π, then they will rephase, and a revival occurs, p 2 with the condition gTrevival( |λ| + 1 − |λ|) = 2π, giving gTrevival = 4π|λ|. Thus, the characteristic timescales are given by: 1 1 |λ| T ' ,T ' ,T ' . (3.19) oscillation g|λ| decay g revival g and the associated behaviour is illustrated in Fig. 3.1. Let us now consider how this behaviour may be approximated ana- lytically. The simplest approach might be to replace the sum over number states by integration, however this would inevitably lose the revivals, which are associated with the discreteness of the sum allowing rephasing. Thus, it is necessary to take account of the discreteness of the sum, and hence the difference between a sum and an integral. For this, we make use of Poisson summation which is based on the result: X X X X Z δ(x − m) = e2πirm ⇒ f(m) = dxe2πirxf(x). (3.20) m r m r This allows us to replace the summation by integration, at the cost of having a sum of final results, however in the current context this is very helpful, as the final sum turns out to sum over different revivals. Applying 1 P this formula, we may write Pex ' 2 [1 − r <{Λ(r, |λ|, t)}] where Z dx x2 x x2 Λ(r, |λ|, t) = exp 2πirx − + igt |λ| + − p2π|λ|2 2|λ|2 2|λ| 8|λ|3 (3.21) 3.3. MANY MODE QUANTUM MODEL — IRREVERSIBLE DECAY23

1 1

0.8 0.5

0.6 0 0 2 4 6

0.4

Excitation probability 0.2

0 0 20 40 60 80 100 120 140 160 180 200 Time × g

Figure√ 3.1: Collapse and revival of Rabi oscillations, plotted for λ = 200.

Completing the square yields:

eigt|λ| Z 1 igt gt Λ(r, |λ|, t) = dx exp − 1 + x2 + i 2πr + x p2π|λ|2 2|λ|2 4|λ| 2|λ| " # eigt|λ| |λ|2 (2πr + gt/2|λ|)2 = exp − p2π|λ|2 2 1 + igt/4|λ| " # Z 1 igt 2πr + gt/2|λ|2 × dx exp − 1 + x − i|λ|2 2|λ|2 4|λ| 1 + igt/4|λ| " # eigt|λ| |λ|2 (2πr + gt/2|λ|)2 = exp − . (3.22) p1 + igt/4|λ| 2 1 + igt/4|λ|

From this final expression one may directly read off: the√ oscillation frequency g|λ|; the characteristic time for the first collapse 2 2/g (by taking r = 0 and assuming gt/|λ| is small); the delay between successive revivals 4π|λ|/g; and finally one may also see that the revival at gt = 4πr|λ| has an increased width√ and decreased height compared to the initial collapse, given by a factor 1 + π2r2 as is visible in figure 3.1.

3.3 Many mode quantum model — irreversible decay

So far we have discussed a two level system coupled to a single mode, with light described either classically or quantum mechanically. One important difference between these situations is that when described classically, there was stimulated absorption and stimulated emission (hence the periodic oscillation), but if the classical field had vanished, then there would have been 24 LECTURE 3. JAYNES CUMMINGS MODEL no transition. For the quantum description, spontaneous emission also ex- isted, in√ that the state |0, ↑i hybridises with the state |1, ↓i, due to the +1 in the n + 1 matrix elements. We will now consider the effects of this +1 factor when we couple a single two level system to a continuum of radiation modes. Our Hamiltonian will remain in the rotating wave approximation, but no longer restricted to a single mode of light, and so we have:

X X gk,n h i H = S + ω a† a + a† S + H.c. (3.23) z k k,n k,n 2 k,n − n,k k where ωk = ck and g e · d k,n = √k,n ba . (3.24) 2 2ε0ωkV

Wigner–Weisskopf approach To study spontaneous emission, we start from the state |0, ↑i, in which there are no photons. The subsequent state can be written as: X |ψi = α(t)|0 ↑i + βk,n(t)|1k,n ↓i, (3.25) k where |1k,ni contains a single photon in state k, n. Then, writing the Schr¨odingerequations as:

X gk,n gk,n i∂ α = α + β , i∂ β = − + ω β + α, (3.26) t 2 2 k,n t k,n 2 k k,n 2 k,n

−it/2 ˜ i(/2−ωk)t one may solve these equations by ﬁrst writing α =αe ˜ , βk = βke , ˜ and then formally solving the equation for βk, with the initial condition βk(0) = 0, to give: Z t gk,n 0 β˜ = −i dt0ei(ωk−)t α˜(t0). (3.27) k,n 2 Substituting this into the equation for α gives:

Z t X gk,n 2 0 ∂ α˜ = − dt0 ei(ωk−)(t −t)α˜(t0) t 2 k,n t Z Z dω 0 = − dt0 Γ(ω)ei(ω−)(t −t)α˜(t0) (3.28) 2π where we have deﬁned:

X gk,n 2 Γ(ω) = 2π δ(ω − ω ) (3.29) 2 k,n k,n

Before evaluating Γ, let us find the resultant behaviour, under the assumption that Γ is a smooth function, meaning that Γ does not significantly vary over the range ± Γ, i.e. dΓ/dω 1. In this case (which corresponds to 3.3. MANY MODE QUANTUM MODEL — IRREVERSIBLE DECAY25 the Markov approximation, meaning a flat effective density of states) the integral over ω in Eq. (3.28) becomes a delta function, so that:

Z t Γ() ∂ α˜ = − dt0Γ()δ(t0 − t)˜α(t0) = − α˜(t) (3.30) t 2 Thus, the probability of remaining in the excited state decays exponentially due to spontaneous decay, with a decay rate Γ() for the probability. The approach used in this section is known as the Wigner–Weisskopf approach.

Eﬀective decay rate We have not yet calculated the eﬀective decay rate appearing in the previous expression. Let us now use the form of gk in Eq. (3.24) to evaluate Γ(). Writing the wavevector in polar coordinates, with respect to the dipole matrix element dab pointing along the z axis (see Fig. 3.2), we then have: ZZZ 2 2 V |dab| X Γ() = 2π dφdθ sin θk2dkδ(ck − ) (zˆ · eˆ )2. (3.31) (2π)3 2ε ckV kn 0 n

θ e2

e1 k z θ

Figure 3.2: Directions of polarisation vectors in polar coordinates

From Fig. 3.2 it is clear that zˆ · eˆk2 = 0 and one may see that zˆ · eˆk1 = sin θ. For concreteness, the k direction and polarisations can be parametrised by the orthogonal set:  sin θ cos φ   cos θ cos φ   − sin φ  ˆ k =  sin θ sin φ  , eˆk1 =  cos θ sin φ  , eˆk2 =  cos φ  . cos θ − sin θ 0 (3.32) Putting all the terms together, one ﬁnds: 2 2 Z Z Z 2 3 |dab| 3 ckcdk 1 4|dab| Γ() = 2 dφ dθ sin θ 3 δ(ck − ) = 3 (3.33) (2π) 2ε0 c 4πε0 3c 26 LECTURE 3. JAYNES CUMMINGS MODEL

This formula therefore gives the decay rate of an atom in free space, associated with the strength of its dipole moment and its characteristic energy.

Further reading

The contents of most of this chapter is discussed in most quantum optics books. For example, it is discussed in Scully and Zubairy [10], or Yamamoto and Imamo˘glu[11].

Questions

Question 3.1: Collapse and revival Consider the “linear” model:

H = ωa†a + c†c + g(a†c + c†a). (3.34)

What is its spectrum? If one starts from a coherent state of photons, exp(λa†)|0, 0i, what is the expectation of occupation of the “atomic” mode c after time t. Explain why there is no collapse. Consider now the model [10, Sec. 6.1]: √ † ! 1 ga a a † H = √ + ω0a a. (3.35) 2 g a†aa† −

Once again, ﬁnd the time evolution of the probability of being in an excited state, and thus ﬁnd the timescales for oscillation, collapse and revival in this case. [Note that this problem is exactly solvable, no approximations should be required].

Question 3.2: Noisy classical driving Let us consider the problem in Eq. (3.1) perturbatively; i.e. to leading order in gλ; for non-degenerate perturbation theory to be valid here, we must for the moment assume we are away from resonance. Show that in ﬁrst order time-dependent perturbation theory, the probability of being in the excited state initially grows as:

Z gλ P (T ) = T dτX(t + τ)X∗(t),X(t) = ei(ω0−)t. (3.36) ex 2

[Hint: Find the standard time-dependent perturbation result for the amplitude in the excited state, then rewrite the amplitude squared to bring out a factor of time delay T .] Now, consider the case where in addition to the desired driving there is some random ﬁeld, eiω0t → eiω0t+η(t). If η(t) is a Gaussian random variable 0 0 with hη(t)i = 0, and hη(t)η(t )i = Γ2δ(t − t ), show that the transition rate now becomes 2 2 Pex(T ) Γ2 g |λ| W = = 2 2 (3.37) T 2 ( − ω0) + Γ2 3.A. FURTHER PROPERTIES OF COLLAPSE AND REVIVAL 27

3.A Further properties of collapse and revival

When considering collapse and revivals of the Jaynes-Cummings oscillations, we discussed only the probability of finding the atom in an excited state. We can however understand more of where this collapse and revival originates from by considering the entanglement between the two- level system and the photon field. The entanglement is defined in terms of S = −Tr(ρT LS ln ρT LS), where ρT LS is the reduced density matrix of the two level system: X ρT LS = hn|ψihψ|ni, (3.38) n summing over e.g. number states of the photon field. Using our previous results for Rabi oscillations, the full state of the photon field can be written as: X X |ψ(t)i = αn(t)|n − 1, ↑i + βn(t)|n, ↓i. (3.39) n n with (restricted to the resonant case): n √ α (t) λ 2 −i sin (g nt/2) n = √ e−|λ| /2 √ . (3.40) βn(t) n! cos (g nt/2) In terms of these, the reduced density matrix will have the form: 2 ∗ X |αn| αn+1βn 1/2 + ZX + iY ρT LS = ∗ 2 = . (3.41) β αn+1 |βn| X − iY 1/2 − Z n n For a two dimensional density matrix, since the trace must be unity, by normalisation, the eigenvalues (and hence the entanglement entropy) are entirely determined by the determinant. In terms of the above parametrisation, the determinant is 1/4−R2 where R2 = X2+Y 2+Z2. If |R| = 1/2, the determinant is zero, and so the eigenvalues are 1, 0, i.e. this is a pure state, with no entanglement. If |R| = 0, then the determinant is 1/4, and the eigenvalues are 1/2, 1/2, with maximum entanglement. Thus, by plotting 1/4 − R2 we have a non-linear parametrisation of the entropy. The vector R is the Bloch vector parametrisation of the density matrix, something we will make much use of in subsequent lectures. From the form of the diagonal matrix elements, and the sum in Eq. (3.15) we have that 1/2 + Z = Pex and so ( " 2 #) 1 X eigt|λ| |λ|2 (2πr + gt/2|λ|) Z = − < exp − . (3.42) 2 p 2 1 + igt/4|λ| r 1 + igt/4|λ| This is non-zero only near the revivals, so if we consider what happens away from the revivals (which is the majority of possible times), then we have |R|2 = X2 + Y 2, which can be written as:

2 λ 2 X 2n −|λ| |R| = |λ| n!e √ n 2 n + 1 2 gt √ √ gt √ √ × sin ( n + n + 1) + sin ( n − n + 1) (3.43) 2 2 28 LECTURE 3. JAYNES CUMMINGS MODEL

In this expression, the sum and difference sine terms can be treated sepa- √ rately. The first term, depending approximately on sin(gt n) will behave very similarly to the excitation probability, with the only difference being this is the imaginary part, rather than the real part, of the complex expression. Thus, once again, away from revivals, this term is small. The last term is however different. Expanding for large n, one has:

X 2 λ gt |R| ' |λ|2nn!e−|λ| √ sin √ ) 4 n n 2 n + 1 2 2 1 gt g t ' = exp i − , (3.44) 2 4|λ| 8|λ|4 where the last line replace the sum by an integral (not worrying about revivals of this discrete sum). This looks superficially similar to the excitation probability, but the timescales are different — the oscillation timescale here is t ∼ 4|λ|/g; this is the same timescale as the revivals of the excitation probability. Such a result makes sense — collapse of the excitation probability occurred because of dephasing, i.e. decoherence, arising from entanglement between the two level system and the photon field. The revival timescale is thus connected to the period of the entanglement oscillations.

1 0.9 0.8 0.7 0.6 0.5 1 0.4 0.3 0.8

Entanglement entropy 0.2 0.1 0.6 180 190 200 0 1

0.5 Excitation probability

0 0 50 100 150 200 250 300 350 400 Time Figure 3.3: Oscillations of the entanglement (top panel) and its connection to collapse and revivals of the excitation probability (bottom panel) in the case of λ = 30. Lecture 4

Two level systems coupled to many modes: Density matrix description

The aim of this lecture is to repeat the physics discussed at the end of the last lecture — i.e. the relaxation of a two-level system coupled to a continuum of radiation modes — but to present it in the context of the density matrix of the two-level system. This framework will then allow us to consider other relaxation and decoherence processes, such as those arising due to noise sources, or inhomogeneous broadening in an ensemble of atoms. We will then illustrate the use of this approach by discussing the behaviour of a two-level system illuminated by a coherent state of light, but able to radiate into a continuum of modes. In this lecture we will study the steady state of this system, and in subsequent lectures we will look at the spectrum of emitted light, which will require knowing more about the full time-evolution of the system. The formalism we develop here will be useful also when we later consider the laser, and its more quantum mechanical variants of single atom lasers and micromasers.

4.1 Density matrix equation for relaxation of two-level system

We begin by considering again the problem of a two-level system coupled to a continuum of radiation modes, which in the previous lecture we solved via the Wigner-Weisskopf approach, of eliminating the Schrödingerequations for the state of the radiation field. Let us now apply the same idea in a more formal scheme, by eliminating the the behaviour of the continuum modes in order to write a closed equation for the density matrix of the system. If we define the system density matrix as the result of tracing over all “reservoir” degrees of freedom, we can write an equation of motion for this quantity as: d ρ = −iTr {[H , ρ ]} . (4.1) dt S R SR SR

29 30 LECTURE 4. DENSITY MATRICES FOR 2 LEVEL SYSTEMS in which S means state of the system, R the reservoir (in this case, the continuum of photon modes), and TrR traces over the state of the reservoir. After removing the free time evolution of the reservoir and system ﬁelds, the remaining system-reservoir Hamiltonian can be written in the interaction picture as:

X gk H = σ−a† ei(ωk−)t + σ+a ei(−ωk)t . (4.2) SR 2 k k k

In general Eq. (4.1) will be complicated to solve, as interactions between the system and reservoir lead to entanglement, and so the full density matrix develops correlations. This means that the time evolution of the system would depend on such correlations, and thus on the history of system reservoir interactions. However, the same assumptions as in the previous lecture will allow us to make a Markov approximation, meaning that the evolution of the system depends only on its current state — i.e. neglecting memory eﬀects of the interaction. For a general interaction, there are a number of approximations which are required in order to reduce the above equation to a form that is straightforwardly soluble.

• We assume the interaction is weak, so ρSR = ρS ⊗ ρR + δρSR, where δρSR 'O(gk). Since ρS = TrR(ρSR), it is clear that TrR(δρSR) = 0, however the result of TrR(HSRδρSR) may be non-zero. Such correlations will be assumed to be small, but must be non-zero in order for there to be any inﬂuence of the bath on the system. In order to take account of these small correlation terms, generated by the coupling, we use the formal solution of the density matrix equation R t 0 0 0 ρSR(t) = −i dt [HSR(t ), ρSR(t )] to write:

Z t d 0 0 0 ρS(t) = −TrR dt HSR(t), HSR(t ), ρSR(t ) . (4.3) dt 0 We may now make the assumption of small correlations by assuming 0 0 0 that ρSR(t ) = ρS(t )⊗ρR(t ), and that only the correlations generated at linear order need be considered.

• We further assume the reservoir to be large compared to the system, 0 and so unaﬀected by the evolution, thus ρR(t ) = ρR(0). Along with the previous step, this is the Born approximation.

• Finally, if the spectrum of the bath is dense (i.e. the spacing of energy levels is small), then the trace over bath modes of the factors eiωkt will lead to delta functions in times. This in eﬀect means that we 0 may replace ρS(t ) = ρS(t). This is the Markov approximation.

One thus has the ﬁnal equation:

Z t d 0 0 ρS(t) = −TrR dt HSR(t), HSR(t ), ρS(t) ⊗ ρR(0) . (4.4) dt 0 4.1. DENSITY MATRIX EQUATION FOR RELAXATION OF TWO-LEVEL SYSTEM 31

To evaluate the trace over the reservoir states, we need the trace of the various possible combinations of reservoir operators coming from HSR. † These will involve pairs of operators ak, ak. Using cyclic permutations, these trace terms can be written as thermal average hXi = TrR(ρRX). For bulk photon modes, the relevant expectations are:

D E D † † E D † E D † E akak0 = akak0 = 0, akak0 = δkk0 nk, akak0 = δkk0 (nk + 1). (4.5) In these expressions we have assumed nk 6= 0, i.e. we have allowed for a thermal occupation of the photon modes. The results of the previous lecture assumed an initial vacuum state, and thus nk = 0. With these expectations, we can then write the equation of motion for 0 the density matrix explicitly, writing ξk = exp[i(ωk − )(t − t )], to give: Z t d X gk 2 ρ (t) = − dt0 σ−σ+n ξ + σ+σ−(n + 1)ξ∗ ρ dt S 2 k k k k s k 0 − + + − ∗ − σ ρσ (nk + 1) + σ ρσ nk (ξk + ξk) − + ∗ + − +ρs σ σ nkξk + σ σ (nk + 1)ξk . (4.6) In this expression, the first term in parentheses comes from the expres- 0 sion TrR [HSR(t)HSR(t )ρSR], the second comes from the conjugate pair 0 0 ∗ TrR [HSR(t)ρSRHSR(t )] and t ↔ t [hence (ξk + ξk)], and the final term 0 comes from TrR [ρSRHSR(t )HSR(t)]. At this point, we can now simplify the above expression by using the assumed flat density of states of the photon modes, so that just as in the Wigner-Weisskopf approach, we may write:

X gk 2 X gk 2 ξ = Γδ(t − t0), n ξ = Γ¯nδ(t − t0). (4.7) 2 k 2 k k k k Here we have assumed that not only is the density of states flat, but that also nk is sufficiently flat, that we might write an averaged occupationn ¯. Such an approximation is only valid at high enough temperatures — we will discuss this in some detail in a few lectures time. d Γ ρ (t) = − n¯ σ−σ+ρ − 2σ+ρ σ− + ρ σ−σ+ dt S 2 S S S Γ − (¯n + 1) σ+σ−ρ − 2σ−ρ σ+ + ρ σ+σ− (4.8) 2 S S S The factor of 1/2 here comes from the regularised half integral of the delta function. The two lines in this expression correspond to stimulated absorption (which exists only ifn ¯ > 0), and leads to excitation of the two-level system, and to emission (both stimulated and spontaneous). Ifn ¯ = 0, only emission survives, and the results of the previous lecture are recovered for the relaxation to equilibrium. One can see the behaviour from this equation most clearly by writing the equations for the diagonal components: Γ Γ ∂ ρ = − n¯ (−2ρ ) − (¯n + 1) (2ρ ) = −∂ ρ (4.9) t ↑↑ 2 ↓↓ 2 ↑↑ t ↓↓ 32 LECTURE 4. DENSITY MATRICES FOR 2 LEVEL SYSTEMS

Then, using ρ↓↓ = 1 − ρ↑↑, one may ﬁnd the steady state probability of excitation is given by: n¯ 0 = Γ [¯n(1 − ρ ) − (¯n + 1)ρ ] → ρ = . (4.10) ↑↑ ↑↑ ↑↑ 2¯n + 1 This formula is as expected, such that if one uses the Bose-Einstein occupation function for the photons,n ¯ = [exp(β)−1]−1, the excitation probability of the two-level system is the thermal equilibrium expression.

Equations of motion with coherent driving As a slightly less trivial example of the eﬀect of the bath, let us now turn to the combination of coherent driving and decay. We will assume from hereon that the reservoir photon modes are empty 1. The evolution of the density matrix is then controlled by:

Γ ∂ ρ = −i [H, ρ] − σ−σ+ρ − 2σ+ρ σ− + ρ σ−σ+ (4.11) t 2 S S S where H describes the coherent, Hamiltonian dynamics, which we will take to be given by: gα H = σz + e−iωtσ+ + H.c.. (4.12) 2 One may then write out the equations of motion for each component explicitly, giving h gα gα i ∂ ρ = −i ρ + e−iωtρ − ρ eiωt − ρ − Γρ , (4.13) t ↑↑ 2 ↑↑ 2 ↓↑ ↑↓ 2 ↑↑ 2 ↑↑ h gα gα i Γ ∂ ρ = −i ρ + e−iωtρ − ρ e−iωt − ρ − − ρ . t ↑↓ 2 ↑↓ 2 ↓↓ ↑↑ 2 ↑↓ 2 2 ↑↓ (4.14)

These two equations are sufficient, since the unit trace property of a density matrix implies ρ↓↓ = 1 − ρ↑↑, and the off diagonal elements are complex conjugate of one another by the Hermitian structure of the density matrix. To solve these equations, it is convenient to first go to a rotating frame, −iωt so that one writes ρ↑↓ = P e , and secondly to introduce the inversion, Z = (ρ↑↑ − ρ↓↓)/2. In terms of these variables, one has:

gα 1 ∂ Z = −i (P ∗ − P ) − Γ Z + (4.15) t 2 2 gα Γ ∂ P = i(ω − )P + i 2Z − P. (4.16) t 2 2 To further simplify, we may separate the real and imaginary parts of P = X + iY , and write ∆ = ω − for the detuning of the optical pump, so that one has the three coupled equations:

1This limit is frequently relevant in cavity quantum Electrodynamics experiments, since the conﬁned photon modes start at energies of the order of 1eV∼ 104K, so the thermal population of such photon modes is negligible. 4.2. DEPHASING IN ADDITION TO RELAXATION 33

Γ ∂ X = −∆Y − X (4.17) t 2 Γ ∂ Y = +∆X + gαZ − Y (4.18) t 2 1 ∂ Z = −gαY − Γ Z + . (4.19) t 2

These are Bloch equations for the Bloch vector parametrisation of the density matrix (in the rotating frame):

1 2 + ZX − iY ρ = 1 . (4.20) X + iY 2 − Z The three terms, ∆, gα, Γ correspond to: rotation about the Z axis (i.e. phase evolution); rotation about the X axis (i.e. excitation — nb the observation that the rotation rate is gα, so a duration gαt = π would lead to the inverted state); and Γ causes the length of the Bloch vector to shrink. As noted in the appendix to lecture 3, for a two level system, the eigenvalues are directly given by the determinant of the density matrix. Hence, the entanglement entropy, describing the decoherence of the system due to its entanglement with the reservoir, depends monotonically on the length of the Bloch vector, X2 + Y 2 + Z2. Before solving Eq. (4.17–4.19), we will ﬁrst consider other kinds of dephasing that may aﬀect the density matrix evolution, leading to a gen- eralised version of these equations, allowing for both relaxation and pure dephasing.

4.2 Dephasing in addition to relaxation

In addition to relaxation, where excitations are transfered from the system to the modes of the environment, pure dephasing is also possible, in which the populations are unaﬀected but their coherence is reduced. In considering this kind of dephasing, it is useful to also broaden our view from considering a single two level system, to describing measurements on an ensemble of many two level systems. For the moment we will assume that even if there are many two level systems, they all act independently. As such, the expectation of any measurement performed on this ensemble can be given by: X X hXi = T r(ρiX) = T r(¯ρX), whereρ ¯ = ρi. (4.21) i i We thus have two types of decoherence one may consider:

• Broadening within a two-level system, coming from time dependent shifts of energies (and possibly also coupling strengths). A variety of sources of such terms exist; examples include collisional broadening of gaseous atoms, where the rare events in which atoms approach 34 LECTURE 4. DENSITY MATRICES FOR 2 LEVEL SYSTEMS

each other lead to shifts of atomic energies. In solid state contexts, motions of charges nearby the artiﬁcial atom can also lead to time dependent noise.

• Inhomogeneous broadening, where the parameters of each two-level system vary, and so off-diagonal matrix elements, involving time dependent coherences, are washed out inρ ¯. This is particularly an issue in solid state experiments, where the two-level systems may involves the dimensions of a fabricated or self-assembled artificial atom, and these dimensions may vary between different systems.

To describe both such eﬀects together, we consider adding noise,

→ + η(t). (4.22)

This noise will then lead to a decay of the oﬀ diagonal correlations, via:

Z t P (t) → P (t) exp i η(t0)dt0 (4.23) 0

We will take hη(t)i = 0, and two distinct limits for the correlations of η.

Fast noise limit In order to extract a tractable model of time-dependent noise, the simplest limit to consider is that η(t) has a white noise spectrum, i.e. it is Gaussian correlated with: 0 0 hη(t)η(t )i = 2γ0δ(t − t ). (4.24) This assumption is equivalent to assuming that the noise is “fast”, i.e. that any correlation timescale it does possess is suﬃciently short compared to the dynamics of the system that it may be neglected. With this assumption of instantaneous Gaussian correlations, one may then expand the exponential in Eq. (4.23) and write:

Z t 1 ZZ t P (t) →P (t) 1 + i dt0η(t0) − dt0dt00η(t0)η(t0) + ... 0 2! 0 " # 2γ Z t (2γ )2 Z t 2 =P (t) 1 − 0 dt0 + 0 3 dt0 + ... = P (t)e−γ0t 2! 0 4! 0 (4.25) with the combinatoric factors coming from the number of possible pairings in the Wick decomposition of hηni. This noise thus leads to an enhanced decoherence rate for the oﬀ diagonal terms, so that Γ/2 should be replaced by 1/T2 = (Γ/2) + γ0 in the density matrix equations. One may equivalently deﬁne the relaxation time 1/T1 = Γ. In general T2 < T1, as dephasing is faster than relaxation, however the factors of two in the above mean that the strict inequality is T2 < 2T1, with equality holding when there is only relaxation. 4.2. DEPHASING IN ADDITION TO RELAXATION 35

The Bloch equations are thus given by:

1 ∂tX = −∆Y − X (4.26) T2 1 ∂tY = +∆X + gαZ − Y (4.27) T2 1 1 ∂tZ = −gαY − Z + . (4.28) T1 2

Static limit (inhomogeneous broadening) To describe inhomogeneous broadening, η should represent the system- dependent variation of energies, and should have no time dependence. The same behaviour is also relevant if time-dependent noise terms vary slowly on the timescale of an experiment. (For example, slow noise may arise from a nearby impurity have multiple stable charging states, so that the charge environment can take multiple diﬀerent values; if the charging and discharging of this impurity is slow compared to the experiment, this energy shift will eﬀectively give inhomogeneous broadening when averaged over multiple experimental shots.) For this case η is static, but randomly distributed, and Eq. (4.23) becomes: Z P (t) → P (t) dηp(η)eiηt. (4.29)

If we consider a Lorentzian probability distribution for η with width γ1, i.e. 2 2 p(η) = 2γ1/(γ1 + η ), then one ﬁnds P (t) → P (t) exp(−γ1|t|). If one has both noise terms as in the previous section, and inhomoge- ∗ neous broadening, one may then distinguish T2 as deﬁned above, and T2 , where 1/T2 = (Γ/2) + γ0 + γ1.

∗ Distinguishing T2 and T2

Since both γ0 and γ1 lead to decay of the coherence functions, a simple ∗ experiment measuring coherence will see the lifetime T2 . However, the dynamics of each individual two level system remains coherent for the longer timescale T2. Such extra coherence can easily be seen in any suﬃciently ∗ non-linear measurement. An example of how the diﬀerence of T2 and T2 can be measured is given by photon echo, illustrated in Fig. 4.1.

Z Z Z Z g α t= π

X X X X

Y Y Y Y

g α t= π /2

Figure 4.1: Cartoon of photon-echo experiment, which removes dephasing due to inhomogeneous broadening, leaving only true dephasing rate to reduce intensity of revival. 36 LECTURE 4. DENSITY MATRICES FOR 2 LEVEL SYSTEMS

From the Bloch equations, Eq. (4.26–4.28), it is clear that a resonant pulse with duration gαt = π/2 brings the Bloch vector to the equator. From there, inhomogeneous broadening means that (in a rotating frame at the mean frequency) the Bloch vectors for each individual two-level system start to spread out. However, by applying a second pulse with gαt = π, a π rotation about the X axis means that whichever Bloch vector spread at the fastest rate is now furthest from the Y axis, and the subsequent evolution sees the vectors re-converge. At the ﬁnal time, the coherence of the resulting pulse will have been reduced by γ0 and Γ, but γ1 will have had no eﬀect.

4.3 Power broadening of absorption

Having developed this formalism, we may now use this to re-examine the question we started the previous lecture with — what is the eﬀect of shining classical light on a two-level atom, but now accounting for decay. Whereas in the previous lecture, the absence of decay meant that the result was Rabi oscillations, with decay a steady state is eventually found. Looking for steady state solutions of Eq. (4.26–4.28) one ﬁnds from the equations for X and Z that: −1 1 ∆T2 1 Y = Z + ,X = −∆T2Y = Z + . (4.30) gαT1 2 gαT1 2

Substituting these into the equation for Y , and recalling Pex = Z + 1/2, gives: ∆T2 1 1 gα 0 = Pex ∆ + + gα − (4.31) gαT1 T2 gαT1 2 After re-arranging, one then ﬁnds:

2 2 1 T1T2(gα) 1 (T1/T2)(gα) Pex = 2 2 = 2 2 2 (4.32) 2 (T2∆) + 1 + T1T2(gα) 2 ∆ + [1 + T1T2(gα) ] /T2 The addition of damping to the equations of motion has thus had a number of important consequences:

• It is responsible for a steady state existing at all (but note that this steady state is in a frame rotating at the pump frequency ω, so the physical coherence is time dependent.)

• It gives a ﬁnite width to the resonance, even at weak pump powers — in the Hamiltonian case, as gα → 0, the width of the resonance peak vanished.

• It modiﬁes the overall amplitude at resonance, by a factor depending on the power, such that at gα → 0, there is no response.

One feature of this absorption probability formula that can be remarked on is that the linewidth depends on the intensity of radiation. This is a consequence of the nonlinearity implicit in a two-level system. Such 4.4. FURTHER READING 37 dependence is not so surprising given that in the Hamiltonian dynamics, the resonance width already depended on the ﬁeld strength. One may note that if one considered an harmonic atomic spectrum, rather than a two-level atom, no such broadening would be seen (see question 4.2).

0.5

0.4

0.3 ex P 0.2

0.1

0 -30 -20 -10 0 10 20 30 ∆ Figure 4.2: Power broadening of absorption — probability of excitation vs atom-photon detuning for increasing ﬁeld strength.

4.4 Further reading

Once again, the contents of this chapter can be found in most standard quantum optics books, e.g.[10, 12]. The discussion of the resonance ﬂuo- rescence problem in particular can be found in chapter 4 of Meystre and Sargent III [12].

Questions

Question 4.1: Fast noise via density matrix equations As another way to consider fast noise terms, consider interaction between the two-level system and a bath of radiation modes given by the interaction picture coupling: X z −iωnt † iωnt HSR = σ ζn Bne + Bne (4.33) n where Bn are the quantised modes of a ﬁeld whose ﬁeld strength shifts the energies of the two-level system.

4.1.(a) By making the Born-Markov approximation, show that this leads to a density matrix equation of the form: ρ ∂ ρ| = −γ − 2σzρσz , (4.34) t noise 0 2 and determine how γ0 depends on the temperature of the reservoir if the reservoir follows a thermal Bose-Einstein distribution. 38 LECTURE 4. DENSITY MATRICES FOR 2 LEVEL SYSTEMS

4.1.(b) Show that this describes pure dephasing, i.e. it has no eﬀect on the diagonal elements of the density matrix, and leads to a decay of the oﬀ-diagonal terms ∂tρ↑↓ = −γ0ρ↑↓.

Question 4.2: Absence of power broadening for an harmonic oscillator † Consider an harmonic oscillator mode Hsys = b b, coupled to a continuum of radiation modes via the interaction picture Hamiltonian:

X gk H = ba† ei(ωk−)t + b†a ei(−ωk)t . (4.35) SR 2 k k k

4.2.(a) Show that (for an empty bath) this leads to the density matrix equation of motion: Γ ∂ ρ = − b†bρ − 2bρb† + ρb†b (4.36) t 2 4.2.(b) Now consider adding the Hamiltonian dynamics describing pumping by a coherent ﬁeld, with H = (gα/2)(b†e−iωt + H.c.) + b†b. Determine the density matrix equation equation of motion in the Fock basis (i.e. number state basis).

4.2.(c) Show that the ansatz:

1 −igα/2 n igα/2 m ρnm ∝ √ (4.37) n!m! −i∆ + Γ/2 i∆ + Γ/2 satisﬁes this equation of motion.

4.2.(d) From the normalised density matrix, ﬁnd the average excitation probability, and show that there is no power broadening. Lecture 5

Resonance Fluorescence

At the end of the last lecture, we found the steady state excitation probability for a two-level system pumped by a coherent light ﬁeld. This steady state emerged due to the competition between coherent pumping, and incoherent decay into the continuum of radiation modes. The aim of this lecture is to discuss the spectrum of this ﬂuorescence, when the atom is driven near resonance. This is intended both as an illustration of applying the density matrix equation of motion approach to more complex problems, and also to reveal further aspects of the behaviour of a two-level system.

5.1 Spectrum of emission into a reservoir

Our aim is to calculate the spectrum of the emission into the continuum of photon modes. Formally, the spectrum of radiation is given (via the Wiener-Khintchine theorem, applicable to a stationary process, which ours is) by:

Z ∞ Z ∞ I(ν) = eiνthE+(t)E−(0)i = 2< eiνthE+(t)E−(0)i (5.1) −∞ 0 where we have divided:

+ − − X E(t) = E (t) + E (t),E (t) = Ekak,n(t)en,k, (5.2) k p with Ek = ωk/2ε0V . The ﬁrst challenge is to write the time dependent reservoir operators in terms of system operators. If we work in the Heisen- berg picture, then this is straightforward, the interaction Hamiltonian was:

X gk H = σ−a† ei(ωk−)t + σ+a ei(−ωk)t . (5.3) SR 2 k k k and so the solution to the Heisenberg equation for ak(t) can be written as:

Z t gk 0 a (t) = dt0ei(ωk−)(t −t)σ−(t0) + a (−∞). (5.4) k 2 k

39 40 LECTURE 5. RESONANCE FLUORESCENCE

We will neglect second term ak(−∞), since we assume the state at t = −∞ to be a vacuum, i.e. the only photons are those coming from the atom. In this case we may write:

Z t X Ekgk 0 E−(t) = dt0 ei(ωk−)(t −t)σ−(t0)e , (5.5) 2 n,k k

Because (at least near resonance) Ek ∝ gk the sum in the above expression is Markovian as in the previous lecture, and so we may write E−(t) ∝ σ−(t), hence up to an overall constant, we have: Z ∞ I(ν) ∝ 2< eiνthσ+(t)σ−(0)i . (5.6) 0 This reduces the problem to that of finding a two-time correlation function of the open quantum system. This is however something we have not yet calculated — the density matrix completely describes single time correlation functions, but in general more knowledge is required to find the correlation of operators at two different times. However, for Markovian baths it turns out that it is possible to relate this two-time correlation to the density matrix evolution via the quantum regression theorem.

5.2 Quantum regression “theorem”

The quantum regression theorem can be stated in terms of the time evolution of single-time correlation functions, governed by: X hB(t + τ)i = αi(τ)hBi(t)i, (5.7) i where Bi(t) is some complete set of operators, and the functions αi(τ) solve the averaged equations of motion for hBi(t)i. Then, the theorem states that one can write: X hA(t)B(t + τ)C(t)i = αi(τ)hA(t)Bi(t)C(t)i, (5.8) i

Note that ordering is important, as A, Bi,C are non-commuting operators. In order to prove the theorem, it is ﬁrst necessary to state more explore further what is implied by the Markovian approximation.

Signiﬁcance of the Markovian approximation Formally stated, a Markovian system is one where the future evolution is governed by the current state, and not by any history of the system. In the current context, current state means the state of the system, not including the baths. Thus, the assumption is that one can write an equation of motion for the system density matrix that depends only on the current value of the system density matrix. Therefore, the baths must have no memory of previous states of the system. 5.2. QUANTUM REGRESSION “THEOREM” 41

Generically, without the Markovian approximation, the equation of motion of the system density matrix could be written as Z t dρS(t) 0 0 0 = −i[HS, ρS(t)] + dt Γ(t − t )ρS(t ). (5.9) dt −∞ The term Γ(t−t0) describes how the state of the bath influences the system at time t, and depends on the state of the system at the earlier time t0. The Markovian approximation is that Γ(t − t0) = Γδ(t − t0). If one describes the bath by a sum over many independent modes, these delta correlated response functions imply a dense spectrum of bath modes; this is what was used in Eq. (4.7). There is another consequence of the Markovian limit, which was also used earlier in Eq. (4.4). The full evolution really depends on ρSR; validity of the Markovian approximation then requires that the state of the system is sufficiently described by ρS = TrRρSR. This means that correlations between the system and the reservoir must be unimportant, and so it is sufficient to write ρSR(t) = ρS(t) ⊗ ρR(t). It is this statement, as we see next, that implies the quantum regression theorem. Under what physical conditions is the Markovian approximation a good description of a real system? Firstly, to have truly delta correlated noise requires a flat spectrum of the baths, as we have already assumed. However, a flat spectrum is only required for the bath modes that couple almost resonantly to the system (i.e. |ωj − ω| ≤ Γ). If one considers a reservoir which is not in its vacuum state — i.e. with some thermal occupation — then a Markovian description of the system requires that not only is the density of states flat, but that we may also approximate:

X gk 2 n ei(ωk−ω)t =n ¯Γδ(t). k 2 k i.e. we require that nk varies slowly for over the range of k for which |ωj − ω| ≤ Γ; e.g. for a thermal distribution, we require T Γ. This is clearly always an approximation; the consequences of this approximation will be investigated further in the next lecture.

Proof of regression theorem The Markovian approximation implies that the evolution of single-time expectations can be written as follows, in terms of the unitary evolution U(τ) of system and reservoir:

† hB(t + τ)i = TrSTrR U (τ)B(t)U(τ)ρS(t) ⊗ ρR(t)

h † i = TrS B(t)TrR U(τ)ρS(t) ⊗ ρR(t)U (τ) h i = TrS B(t)TrR ρS(t + τ) ⊗ ρR(t + τ) . (5.10)

Here we made use of the cyclic property of the trace, and the relation between evolution of density matrices and that of Heisenberg operators. In 42 LECTURE 5. RESONANCE FLUORESCENCE an analogous way, the two-time expectation evolves according to:

† hA(t)B(t + τ)C(t)i = TrSTrR A(t)U (τ)B(t)U(τ)C(t)ρS(t) ⊗ ρR(t)

h † i = TrS B(t)TrR U(τ)[C(t)ρS(t)A(t)] ⊗ ρR(t)U (τ) h i = TrS B(t)TrR [CρSA](t + τ) ⊗ ρR(t + τ) . (5.11)

Thus, the only diﬀerence between the right hand sides of Eq. (5.10) and Eq. (5.11) is to replace the initial system density matrix ρS with the product CρSA. However, in the Markovian approximation, we know that Eq. (5.10) can be solved using Eq. (5.7), for any initial density matrix. Thus, one can use this solution to give

X hA(t)B(t + τ)C(t)i = αi(τ)hBi(t)i ρ (t)→C(t)ρ (t)A(t) i s S X = αi(τ)TrS Bi(t)C(t)ρS(t)A(t) i X = αi(τ)TrS A(t)Bi(t)C(t)ρS(t) (5.12) i which is the desired result in Eq. (5.12). Thus, to find a two-time correlation in practise, the scheme is this: find the general time-dependent solution of one of the operators, and how it depends on initial expectation values — find the explicit form of Eq. (5.7), and then insert the second operator(s) in these initial expectations.

5.3 Resonance ﬂuorescence spectrum

From the quantum regression theory, we now have that:

Z ∞ iνt + I(ν) ∝ 2< e Tr σ ρ(t) ρ(t=0)=σ−ρ , (5.13) 0 0 i.e. we should solve the density matrix equation of motion to find ρ(t) given − the initial condition σ ρ0, where ρ0 is the equilibrium density matrix. Since the equation of motion for the density matrix is a first order differential equation, ∂tρ = M[ρ], where M is a linear super-operator, we can easily solve by Laplace transforming to give:

− M[ρ(s)] − sρ(s) = −σ ρ0, (5.14) where the right hand side encodes the initial conditions. From the Laplace transform ρ(s), the Fourier transform corresponding to the intensity spectrum is then given by:

I(ν) ∝ 2< Tr σ+ρ(s = −iν + 0) . (5.15) 5.3. RESONANCE FLUORESCENCE SPECTRUM 43

Let us now specify the various parts of this for the case of resonance ﬂuorescence, i.e. ∆ = ω − = 0. WE will work in the rotating frame as in the previous lecture. The steady state condition then becomes:

2 1 T1T2(gα) iT2(gα) ρ0 = 2 2 (5.16) 2[1 + T1T2(gα) ] −iT2(gα) 2 + T1T2(gα)

(making use of X = 0 and Y,Z as calculated in the previous lecture.) The important part of the spectrum however comes from the super- operator in Eq. (5.14). If we write density matrices as a vector:   ρ↑↑ ρ↑↑ ρ↓↑  ρ↓↑  ρ = → ρ =   , (5.17) ρ↑↓ ρ↓↓  ρ↑↓  ρ↓↓ then in this four dimensional space, the super-operator can be written as a 4 × 4 matrix, following Eq. (4.15)   −1/T1 −igα/2 igα/2 0  −igα/2 −1/T2 0 igα/2  M =   . (5.18)  igα/2 0 −1/T2 −igα/2  1/T1 igα/2 −igα/2 0

One can thus ﬁnd the full density matrix as:

  0   + −1  0  I(ν) ∝ −2< (0 0 1 0)[M − (−iν + 0 )1]  2  . (5.19)   T1T2(gα)  +iT2(gα)

When inverting the 4 × 4 matrix, there will in general be four poles. How- ever, because the density matrix evolution is trace-preserving, one of these poles will necessarily be at zero, corresponding to the conserved quantity. The corresponding eigenvalue will be the steady state (the trace-preserving property ensures that the steady state is non-trivial). Since we know that in the steady state there is a non-zero polarisation, the weight of this zero pole will not in general vanish. This means there is a delta-function peak in the spectrum. Such a peak describes the elastic scattering of pump photons into the radiation modes. The remainder of the poles will describe inelastic scattering. To ﬁnd these poles, we should solve:

1 1 1 0 = Det [M + iν] = iν iν − iν − iν − + (gα)2 . T2 T2 T1 (5.20) This has the zero mode as discussed above, and three other poles: s i 1 1 1 1 2 ν = − , ν = −i + ± (gα)2 − − . (5.21) T2 2T1 2T2 2T1 2T2 44 LECTURE 5. RESONANCE FLUORESCENCE

The ﬁrst of these will describe a broadened resonance near elastic scattering (remembering that this is in the rotating frame). The second term describes (at large enough α), a resonance shifted by ±α. When the weights of these poles are calculated, as shown in Fig. 5.1, the central peak has twice the weight of the other two peaks. This characteristic shape is known as the Mollow triplet, and has a simple interpretation.

ω=ε T1=T2=2 gα=10 Intensity

-15 -10 -5 0 5 10 15 ν − ω

Figure 5.1: Spectrum of resonance ﬂuorescence, from Eq. (5.19).

Interpretation: of the Mollow triplet The origin of the Mollow triplet is the effective energy levels of the two- level system in the presence of a coherent field. If one considers in general a two-level system coupled to photons, there are Rabi oscillations. In the frequency domain, these oscillations correspond to a splitting of the eigenstates resulting from hybridising the states with n excitations; i.e. the states √ |n − 1, ↑i, |n, ↓i mix and lead to eigenstates split by g n (if on resonance). If the system can now decay, it will emit an excitation into the continuum, and undergo a transition to a state with one fewer excitation in the system. However, starting from one of the hybridised eigenstates with n excitations, it is possible to undergo a transition to either of the eigenstates with n−1 excitations — non-zero matrix elements connect them all. Thus, in general there are four transition frequencies. In the limit of a strong classical field, all four of these matrix elements have the same strength, and moreover, as indicated in Fig. 5.2, the Rabi splitting becomes almost n independent when n 1, so two of the frequencies will match. This then leads to the characteristic 1:2:1 ratio seen in Figure 5.1.

5.4 Further reading

The discussion of the resonance ﬂuorescence problem in particular can be found in chapter 16 of Meystre and Sargent III [12]. The discussion of the quantum regression theorem in this lecture is however more closely related 5.4. FURTHER READING 45

gα /2 gα /2

ω∼ε

gα /2 gα /2

Figure 5.2: Characteristic behaviour at high excitation number, with n ' |α|2, demonstrating a triplet of transition frequencies, with the intensity ratio 1:2:1. to the density matrix based approach given in Scully and Zubairy [10]; The Onsager-Lax (quantum regression) theorem was introduced by Lax [13].

Lecture 6

Quantum stochastic methods and limitations of the Markovian approximation.

The previous two lectures, have introduced the density matrix equations of motion to describe a system coupled to a bath, and have shown how the Markovian approximation implicit in these equations of motion leads one to the quantum regression theorem. This lecture reviews the foundations of these methods for open systems. Two further formalisms are presented, which can be seen as the stochastic equations of motion for which the probability evolution is that of the density matrix. In the latter of these — the Heisenberg-Langevin equations — the consequences of the approximations required for a Markovian equation of motion are more explicit; these are discussed in terms of the ﬂuctuation dissipation theorem. We will consider just the problem of a system coupled to a single decay bath, which gave us the equation of motion:

d κ h ρ(t) = − n¯ aa†ρ − 2a†ρa + ρaa† dt 2 i + (¯n + 1) a†aρ − 2aρa† + ρa†a . (6.1)

6.1 Quantum jump formalism

Consider evolution under Eq. (6.1) whenn ¯ = 0, i.e. when only decay can occur. Then, after a short time δt, one has: κδt ρ (t + δt) = ρ (t) − a†aρ + ρ a†a + κδtaρ a† + O(δt2) S s 2 s s s κδt κδt √ √ = 1 − a†a ρ 1 − a†a + κδtaρ κδta† + O(δt2). 2 s 2 s (6.2) These two contributions to the density matrix can be interpreted as the conditional density matrices that arise under “no photon loss” and “one photon loss”; they are added together because of the trace over states of

47 48 LECTURE 6. QUANTUM STOCHASTIC METHODS the bath. The probability of these two ﬁnal states can be found from the trace of the conditional density matrices, hence:

† D † E Pone photon loss = Tr κδtaρsa = κδt a a . (6.3)

This is as one intuitively expects; the probability of losing a single photon in [t, t + δt] is given by the rate of loss of each photon multiplied by the total number of photons. Thus, the evolution can be written in terms of the probabilities of transition to the (suitable normalised)  a|ψi  P = κδthψ|a†a|ψi phψ|a†a|ψi |ψi → (6.4) 1 − κδta†a/2 |ψi  P = 1 − κδthψ|a†a|ψi phψ|1 − κδta†a|ψi This quantum jump formalism can be understood as describing the loss process in terms of the environment repeatedly measuring the number of photons in the cavity, and hence decohering the states with diﬀerent numbers of photons.

Example: Photon loss from a single-photon state The above procedure can be illustrated for a particularly simple case: if the initial state contains a single photon then from the density matrix equation: κ ∂ hn + k|ρ|ni = − (2n + k)hn + k|ρ|ni t 2 + κp(n + 1)(n + k + 1)hn + k + 1|ρ|n + 1i (6.5) it is clear that the initial condition ∀n : hn + k|ρ|ni = 0 is preserved except for k = 0, and so the only equations to solve are

∂th1|ρ|1i = −∂th0|ρ|0i = −κh1|ρ|1i. (6.6) so the full solution is:

ρ = e−κt|1ih1| + (1 − e−κt)|0ih0|. (6.7)

In the quantum jump formalism it is easy to reproduce this result; at each time step we have that: ( |0i P = κδt |1i → (6.8) |1i P = 1 − κδt

A more complicated example is given in Question 6.1, for decay of a coherent state. In that case, since a coherent state is unchanged under the loss of a photon, one has the counterintuitive result that it is failure to observe a photon that leads to the density matrix evolving. This is less surprising if one considers that the state has an uncertain number of photons to start with, and failure to observe any photons escape indicates that the wavefunction has been projected by observation onto a state with fewer photons. 6.2. HEISENBERG-LANGEVIN EQUATIONS 49

6.2 Heisenberg-Langevin equations

Just as one can solve a closed quantum-mechanical problem in either the Schr¨odingeror Heisenberg pictures, quantum stochastic approaches can be constructed in either picture. The previous sections generally described a stochastic evolution of the wavefunction, this section instead describes stochastic evolution of the system operators. Starting once again from the same coupled bath and system, and working in the rotating frame as normal, one has the Hamiltonian of Eq. (4.2), X † i(ω−ωk)t † −i(ω−ωk)t HSR = ζk abke + a bke . (4.2) k From this Hamiltonian, the Heisenberg operator equations of motion are:

X −i(ω−ωk)t ia˙ = [a, HSR] = ζkbke (6.9) k ˙ i(ω−ωk)t ibk = [bk,HSR] = ζkae . (6.10) To eliminate the bath degrees of freedom, one can integrate Eq. (6.10), and substitute it in Eq. (6.9). Thus, t Z 0 0 i(ω−ωk)t 0 bk(t) = −iζk dt e a(t ) + bk(0) (6.11) 0 which, substituted into Eq. (6.9), gives:

t " # " # Z 0 0 X 2 i(ω−ωk)(t −t) 0 X −i(ω−ωk)t a˙ = − dt ζk e a(t )−i ζkbk(0)e . (6.12) 0 k k

Then, defining the second term in brackets as Fa(t), and making the same Markovian approximation: X Z dν ζ2F (ω ) → κF (ν) (6.13) k k 2π k as in Eq. (4.7), so that the first term in brackets becomes κδ(t − t0), this equation becomes: 1 a˙ = − κa + F (t). (6.14) 2 a Here Fa(t) is a stochastic operator; it has quantum mechanical commutation relations related to its definition in terms of bk; but bk is a bath operator, and the state of the bath is random — drawn from a thermal ensemble, so bk has different expectations for each realisation of the bath.

Preservation of commutation relations

The operator nature of Fa(t) is apparent if one considers the commutation relations: 0 † 0 X † −i(ω−ωk)t+i(ω−ωk0 )t [Fa(t),Fa (t )] = ζkζk0 [bk, bk0 ]e k,k0 0 X 2 −i(ω−ωk)(t−t ) 0 = ζk e = κδ(t − t ). (6.15) k 50 LECTURE 6. QUANTUM STOCHASTIC METHODS

This non-commuting nature is essential to ensure the preservation of commutation relations in the time-evolution of the operators a(t). The solution of Eq. (6.14) can be written

Z t −κt/2 0 0 −κ(t−t0)/2 a(t) = a(0)e + dt Fa(t )e . (6.16) 0

Using this, one can write:

h i h i a(t), a†(t) = a(0), a†(0) e−κt Z t Z t 0 00 h 0 † 00 i −κt+κ(t0+t00)/2 + dt dt Fa(t ),Fa (t ) e , (6.17) 0 0 in which we have assumed no correlation between the initial state of the system and that of the bath. Then, using [a, a†] = 1 in the initial state, and the commutator in Eq. (6.15) this becomes:

t h i Z 0 a(t), a†(t) = e−κt + dt0κe−κt+κt 0 = e−κt + 1 − e−κt (6.18)

This shows that the existence of the non-commuting stochastic terms was essential to preserve the operator commutation relations.

Finite bath occupation; other correlations To reproduce all the results of the density-matrix formalism, it is necessary also to have expressions for the correlation functions of the stochastic operators Fa(t) as well as their commutator. Given the form for the commutator, all correlation functions can be found from this and the anticommutator,

Dn oE Dn oE 0 † 0 X † −i(ω−ωk)t+i(ω−ωk0 )t Fa(t),Fa (t ) = ζkζk0 bk, bk0 e k,k0 0 X 2 −i(ω−ωk)(t−t ) = ζk (2nk + 1)e k Z dν βν 0 = κ coth e−i(ω−ν)(t−t ) (6.19) 2π 2 βω ≈ κ coth δ(t − t0). (6.20) 2

The last line here makes the additional assumption of T κ, i.e. that the temperature is large compared to the bandwidth, with the bandwidth taken approximately equal to the imaginary width. This assumption is clearly necessary if the entire formalism is to be Markovian (i.e. to depend only on current state, rather than history), and is identical to the approximation made in Eq. (4.7). The validity and limitations of this last approximation are the subject of the next section. 6.3. FLUCTUATION DISSIPATION THEOREM 51

6.3 Fluctuation dissipation theorem and the Markovian approximation

The ﬂuctuation-dissipation theorem relates the correlation function of an operator with its response function. For the photon operators considered here, the correlation function can be deﬁned by:

Z ∞ 1 Dn oE C(Ω) = dτC(τ)eiΩτ ,C(τ) = a(t), a†(t + τ) . (6.21) −∞ 2 The response function can be found by considering the response to a perturbation: X iΩt † δH = fΩe a + H.c. (6.22) Ω The response function α(Ω) is given by

X iΩt ha(t)i = α(Ω)fΩe . (6.23) Ω

The discussion below uses the Heisenberg-Langevin formalism to evaluate the relation between these operators with and without the second Markovian approximation above. The response function is the same in either case, since it is averaged and so does not depend on the stochastic operator Fa(t). In the following it will be necessary to distinguish between the bare operators a(t) and the gauge transformed versionsa ˜(t) = a(t)e−iωt. Thus,

κ X ∂ + ha˜(t)i = i f ei(Ω−ω)t (6.24) t 2 Ω Ω where we have used the Heisenberg equation following from the gauge transformed perturbation Hamiltonian. This yields

i(Ω−ω)t iΩt X ifΩe X fΩe ha(t)i = eiωtha˜(t)i = eiωt = (6.25) i(Ω − ω) + κ/2 (Ω − ω) − iκ/2 Ω Ω hence α(Ω) = 1/[(Ω − ω) − iκ/2]. Next, consider the ﬂuctuation correlation function at long times, so that the initial conditions are not involved. In this case, Eq. (6.16) should become: Z t iωt 0 0 −κ(t−t0)/2 a(t) = e dt Fa(t )e (6.26) −∞ and so:

−iωτ Z t Z t+τ e 0 00 Dn 0 † 00 oE −κ(2t+τ−t0−t00)/2 C(τ) = dt dt Fa(t ),Fa (t ) e (6.27) 2 −∞ −∞ We now have two choices; we may either use the exact relation in Eq. (6.19), or the Markov approximation in Eq. (6.20). 52 LECTURE 6. QUANTUM STOCHASTIC METHODS

Without second Markovian approximation Considering ﬁrst Eq. (6.19), we have:

−iωτ t t+τ ∞ e Z Z Z dν βν 00 0 0 00 C(τ) = dt0 dt00 κ coth ei(ω−ν)(t −t )−κ(2t+τ−t −t )/2 2 −∞ −∞ −∞ 2π 2 e−iωτ Z ∞ dν βν ei(ω−ν)(t+τ−t) = κ coth 2 −∞ 2π 2 [i(ω − ν) + κ/2][−i(ω − ν) + κ/2] 1 Z ∞ dν βν e−iντ = κ coth 2 2 (6.28) 2 −∞ 2π 2 (ω − ν) + (κ/2) hence it is immediately clear that:

βΩ κ/2 βΩ C(Ω) = coth = coth =[α(Ω)] (6.29) 2 (ω − Ω)2 + (κ/2)2 2 which is the required ﬂuctuation dissipation theorem.

With second Markovian approximation In contrast, if we use Eq. (6.20) we have that:

−iωτ t t+τ e Z Z βω 0 00 C(τ) = dt0 dt00κ coth δ(t0 − t00)e−κ(2t+τ−t −t )/2 2 −∞ −∞ 2 −iωτ min[t,t+τ] e βω Z 0 = κ coth dt0e−κ(2t+τ−2t )/2 2 2 −∞ e−iωτ βω = coth e−κ|τ|/2, (6.30) 2 2 from which it follows straightforwardly that

βω C(Ω) = coth =[α(Ω)]. (6.31) 2

Thus, use of the Markovian approximation means that the ﬂuctuation dissipation theory fails, unless T (Ω − ω) > κ. Such a result can also be seen to be a limit for the use of the quantum regression theorem. In that case, the high temperature (classical) answer is known as the Onsager theorem. The quantum extension of the Onsager theorem is an approximation, that derives from the Markov approximation in the dynamics

6.4 Further reading

The topic of Heisenberg-Langevin equations, and stochastic operator equations is discussed brieﬂy in Scully and Zubairy [10], and more fully in Gar- diner and Zoller [14] The implicit dependence of the quantum regression theorem on the use of the Markovian approximation is highlighted in the papers of Ford and O’Connel [15, 16], and the response by Lax [17]. This topic is addressed again in question 6.2. 6.4. FURTHER READING 53

Questions

Question 6.1: Comparison of quantum jump and density matrix equations of motion. Consider an initial coherent state,

† ∗ ρ = exp(−|α|2)eαa |0ih0|eα a (6.32) following the equation of motion of the density matrix withn ¯ = 0, i.e. κ h i ρ˙ = − a†aρ + ρa†a − 2aρa† . (6.33) 2

6.1.(a) Show that this equation of motion can be satisﬁed by the ansatz

6.1.(b) Show that |ψn(t)i = |ψ0(t)i; (i.e. that losing a photon does not change the state).

P 6.1.(c) Find Pn(t), and verify that n Pn(t) = 1, and hence show that ρ(t) = |ψ0(t)i hψ0(t)|.

6.1.(d) Show that at time t, the non-Hermitian evolution leads to

1 |ψ (t)i = exp(−κta†a/2)|ψ (0)i (6.36) 0 N 0 where N is an appropriate normalisation.

6.1.(e) Prove that this result matches that in Eq. (6.34).

Question 6.2: Fluctuation dissipation theorem, and quantum regression This extended question repeats the analysis in the lecture of the limit of validity of the Markov approximation, but for a damped quantum harmonic oscillator. It shows how the above results do not depend on other approximations that were involved, such as rotating wave approximation. Starting from the Hamiltonian,

2 2 " 2 2 # mω p X mjωj pj H = 0 x2 + + (q − x)2 + (6.37) 2 2m 2 j 2m j j 54 LECTURE 6. QUANTUM STOCHASTIC METHODS

One can easily derive the equations of motion:

2 X 2 x˙ = p/m p˙ = −mω0x + mjωj (qj − x) j 2 q˙j = pj/mj p˙j = −mjωj (qj − x)

Eliminating pj and p, one can write:

X mj q¨ = −ω2(q − x);x ¨ = −ω2x + ω2(q − x) (6.38) j j j 0 m j j j

6.2.(a) Show that the equation for qj(t) is satisﬁed by:

" 0 # Z t 0 pj 0 0 0 qj(t) − x(t) = qj cos(ωjt) + sin(ωjt) − dt cos[ωj(t − t )]x ˙(t ) mjωj −∞ (6.39) Substituting this in the equation for x one ﬁnds:

Z t 2 0 0 0 x¨ + ω0x = −2 dt γ(t − t )x ˙(t ) + F (t) (6.40) −∞ where

X mj 2γ(t − t0) = ω2 cos[ω (t − t0)] m j j j " 0 # X mj pj F (t) = ω2 q0 cos(ω t) + sin(ω t) m j j j m ω j j j j

Let us assume a ﬂat spectrum, so γ(t − t0) = γδ(t − t0). This implies

2 Z ∞ X mjωj γ → dω (6.41) m π j 0

0 0 The correlations of F require knowledge of correlations of pj and qj , the initial momenta and coordinates. Using the reasonable assumption of un- 0 0 correlated, diagonal (thermal) states, one has hpj qki = 0, and: 0 0 0 0 hpj pki ωj 2mjωjhqj qki = 2 = δjk (2n(ωj) + 1) = δjk coth . (6.42) mjωj 2kBT

6.2.(b) Using the above correlations and deﬁnition of F (t), show that:

3 1 X mjωj ωj F (t)F (t0) + F (t0)F (t) = cos[ω (t − t0)] coth 2 m2 j 2k T j B Z ∞ γω ω = dω cos[ω(t − t0)] coth . 0 πm 2kBT (6.43) 6.4. FURTHER READING 55

6.2.(c) Show that the general solution to: 2 x¨ + γx˙ + ω0x = F (t), (6.44) p 2 2 can be written (using ω1 = ω0 − γ /4) as: t 0 Z 0 sin[ω (t − t )] x(t) = dt0e−γ(t−t )/2 1 F (t0). (6.45) −∞ ω1

6.2.(d) Thus, show that: 1 C(τ) = hx(t + τ)x(t) + x(t)x(t + τ)i , (6.46) 2 is given by: Z ∞ γω 1 ω C(τ) = dω cos(ωτ) 2 2 2 2 2 coth . (6.47) 0 πm (ω − ω0) + γ ω 2kBT Fourier transforming, this gives the correlation function Z ∞ −iωt γ ω ω C(ω) = dte C(t) = 2 2 2 2 2 coth . (6.48) −∞ (ω − ω0) + γ ω m 2kBT Let us now relate this fluctuation correlation function to the response function. To define the response function α(t), consider the averaged equa- iωt tion motion, responding to a driving force f(t) = e f0: f(t) hxï + γhx˙i + ω2hxi = . (6.49) 0 m

6.2.(e) Show that the response function, hxiω = α(ω)fω is given by: 1 1 α(ω) = 2 2 (6.50) m ω0 − ω + iγω Hence, show one can write the ﬂuctuation-dissipation theorem as: ω C(ω) = = [α(ω)] coth . (6.51) 2kBT Now, let us repeat the above calculations with the Markovian approximation.

6.2.(f) Show that the general time-dependent behaviour of hx(t)i, following Eq. (6.49) is given by: −γt/2 γ hp0i hx(t)i = e hx0i cos(ω1t) − sin(ω1t) + sin(ω1t) (6.52) 2ω1 mω1 Then, using the Markovian approximation for quantum regression, one can ﬁnd C(τ) as deﬁned in Eq. (6.47), by replacing the initial density matrix with ρ(t) → x(t)ρ(t). Assuming equilibrium single-time correlations, one 2 has hx(t)p(t) + p(t)x(t)i = 0, and hx(t) i = (2¯n + 1)/2mω0). Thus, 2¯n + 1 −γτ/2 γ C(τ) = e cos(ωτ) − sin(ω1τ) (6.53) 2mω0 2ω1 56 LECTURE 6. QUANTUM STOCHASTIC METHODS

6.2.(g) Taking the equilibrium population 2¯n + 1 = coth(ω0/2kBT ), show that this yields

2 γ ω ω0 C(ω) = 2 2 2 2 coth (6.54) (ω − ω0) + γ ω mω0 2kBT 6.2.(h) By considering the ﬂuctuation-dissipation theorem, explain why in the classical limit ω0 kBT , an exact regression theorem holds. i.e. in this limit, the equations governing two-time and single-time correlation functions are exactly related by the quantum regression theorem, even without Markovian or weak coupling approximations. Lecture 7

Two-level atom in a cavity: Cavity QED

In the previous few lectures we have discussed a single two-level atom coupled either to a continuum of radiation modes, or considering a cavity, which picks out a particular mode. In this lecture we will consider more carefully the situations in which a cavity will pick out a single mode, by describing how the system changes as one goes from no cavity, via a bad cavity (which modifies the field strength but does not pick out a single mode) to a good cavity, which can pick out a single mode. To study this problem throughout this crossover, we will consider a two level system coupled both to a cavity pseudo-mode (which itself decays), and also coupled to non-cavity modes, providing incoherent decay. We will show how in this model, a bad cavity can lead to either enhanced decay, and a good cavity can lead to periodic exchange of energy between the cavity and the two-level system, i.e. the Rabi oscillations we previously discussed for a perfect cavity. We will then consider some examples of currently studied experimental cavity QED systems, discussing the values of the relevant parameters describing coupling and decay. Before discussing the crossover between bad and good cavities, we first investigate a toy model of a 1D cavity, to put into context the meaning of the pseudo-mode description of the cavity mode.

7.1 The Purcell eﬀect in a 1D model cavity

The aim of this section is to describe how a cavity modiﬁes the rate of decay of a two-level system, coupled to one-dimensional radiation modes. Following the previous discussion of system-reservoir coupling, the decay is characterised by the combination of reservoirs density of states and its coupling to the atom, given by:

X gk 2 Γ(ω) = 2π δ(ω − ck) . (7.1) 2 k

Recalling that gk ∝ Ek, the field strength associated with a single photon, we can find the effect of the cavity on Γ(ω) by determining the spatial

57 58 LECTURE 7. CAVITY QUANTUM ELECTRODYNAMICS proﬁles of the modes of the full system, and inserting these into the above sum.

L/2 a/2

Figure 7.1: Toy model of cavity, length a, in quantisation volume of length L.

We will consider a cavity of size a, embedded in a quantisation volume of length L, where we will take L → ∞ later on. (See figure 7.1). Our aim is to find the spatial profile of the electric field modes, and thus to calculate

X 2 Γ(ω) ∝ δ(ω − ck)|Ek(x = 0)| . (7.2) k Both in the cavity, and outside, the modes will be appropriate combinations of plane waves. Restricting to symmetric solutions (as antisymmetric solutions will vanish at x = 0), and matching the boundary at x = ±L/2, we may write: ( f(x) ≡ A cos(kx) |x| < a E (x) = k 2 . (7.3) k L a g(x) ≡ Bk sin k |x| − 2 |x| > 2 As L → ∞, the normalisation condition will approach the simple result p Bk = 2/L, since the normalisation integral will be dominated by the parts outside the cavity. We now need to find the effect of the imperfect barriers on this mode in order to relate Ak,Bk, and thus find the quantisation condition specifying k, and the mode amplitude, Ek(x = 0) = Ak. As a simple model, let us consider a varying dielectric index, ε = ε0[1 + ηδ(|x| − a/2)], giving the equation: d2 h ai E (x) = −k2 1 + ηδ |x| − E . (7.4) dx2 k 2 k

Let us focus on x > 0; then, given the form of Ek(x) = Θ(a/2 − x)f(x) + Θ(x − a/2)g(x) with f 00 = −k2f and g00 = −k2g, we may write: d2 a a E = −k2E + 2δ x − g0(x) − f 0(x) + δ0 x − [g(x) − f(x)] dx2 k k 2 2 (7.5) Thus, to solve Eq. (7.4) we require that: a a a a ηk2 h a ai f = g , g0 − f 0 = − f + g (7.6) 2 2 2 2 4 2 2 Substituting the forms of f(x), g(x) into this equation, one may eliminate Ak,Bk to give the eigenvalue condition: kL kη ka ka kL cos + cos sin − = 0. (7.7) 2 2 2 2 2 7.1. THE PURCELL EFFECT IN A 1D MODEL CAVITY 59

For finite L, this provides a restriction on the permissible values of k. As L → ∞, the permissible values of k become more dense, such that, in this limit, one may instead consider this equation as depending on two separate values kL and ka; if L a, it is possible to significantly change kL without modifying ka. Thus, we will instead rewrite this equation as a condition that specifies the value of kL given a fixed ka: kL kη ka ka kL kη ka cos 1 + sin cos = sin cos2 . 2 2 2 2 2 2 2 (7.8) Using this condition, one may then eliminate kL, and write Ak in terms of p Bk = 2/L and functions of ka, i.e.:

r 2 1 Ak = − (7.9) L p1 + kη sin(ka/2) cos(ka/2) + (kη/2)2 cos2(ka/2) If we deﬁne Λ = kη/2, then we may use this formula to give the eﬀective decay rate: ω Γ Γ k = = 0 1 2 c 1 + Λ sin(ka) + 2 Λ (1 + cos(ka)) Γ0 4 = , tan θ0 = . Λ2 q Λ2 Λ 1 + 2 + Λ 1 + 4 cos(ka − θ0) (7.10)

25 100

20 10 1

0 15 0.1 0.01

Γ(ω)/Γ 10 7.2 7.6 8

0 0 1 2 3 4 5 6 7 8 ω/ω0

Figure 7.2: Eﬀective decay rate vs frequency for the toy model of a 1D imperfect cavity. Inset compares the exact result, Eq. (7.10) to the Lorentzian approximation of Eq. (7.13)

The form of this function in general is shown in Fig. 7.2. For small k, the eﬀect of the cavity is weak (i.e. kη 1), and so there is little modiﬁcation compared to the result without the cavity. For larger k, there are sharp peaks, which can be described by an approximately Lorentzian form (see the inset of Fig. 7.2). This Lorentzian form can be found by expanding Eq. (7.10) near its peaks, which are at:

k0a = θ0 + (2n + 1)π. (7.11) 60 LECTURE 7. CAVITY QUANTUM ELECTRODYNAMICS

We can then expand Γ(ω) near ω0 = ck0. If Λ 1, and if we may neglect its k dependence, then we ﬁnd:

Γ 1 ' 2 Γ0 Λ2 Λ2 2 2 1 a(ω−ω0) 1 + 2 − 2 1 + Λ2 − Λ4 1 − 2 c 1 ' . (7.12) 1 Λ2a2 2 Λ2 + 4c2 (ω − ω0) Let us introduce the full width half maximum, κ, such that this expression becomes: Γ P (κ/2)2 = 2 2 (7.13) Γ0 (ω − ω0) + (κ/2) which leads to the deﬁnitions: κ 2c 4c = ,P = Λ2 = . (7.14) 2 Λ2a κa

Here Γ0 is the result for a 1D system in the absence of a cavity. The factor P describes the maximum value of Γ/Γ0 at resonance. At resonance, the presence of the cavity enhances the decay rate, and P is the Purcell enhancement factor. It is helpful for comparison to later results to rewrite this as: ω c 2 λ ω P = 4 0 = Q ,Q ≡ 0 (7.15) κ ω0a π a κ where Q is the quality factor of the cavity mode. The enhancement on resonance therefore depends both on the quality factor, and on the mode volume. Away from resonance, the decay rate decreases instead of increasing, since there are no modes with signiﬁcant weight inside the cavity; the minimum value of Γ/Γ0 occurs when k0a = θ0 + 2nπ, and is given by 2 Γ/Γ0 = 1/Λ = 1/P , hence for this 1D case, P describes both the enhancement on resonance, and the reduction oﬀ resonance. As η increases further, the Lorentzian density of states becomes sharply peaked, and so the Markovian approximation for decay of the atom no longer holds, instead one can use:

Z t Z 2 i(−ω)(t0−t) d − 0 − 0 dω Γ0P (κ/2) e σ = − dt σ (t ) 2 2 dt 2π (ω − ω0) + (κ/2) t Γ P κ Z 0 0 = − 0 dt0σ−(t0)ei(−ω)(t −t)−κ(t−t )/2. (7.16) 4

This form describes the behaviour one would see if the two-level system were coupled to another degree of freedom, with frequency ω and decay rate κ/2. Hence, we are led in this limit to introduce the cavity pseudo- mode as an extra dynamical degree of freedom. The density of states implies that the cavity mode cannot be treated as a Markovian system, it has a non-negligible memory time 1/κ, and so we must consider its dynamics. 7.2. WEAK TO STRONG COUPLING VIA DENSITY MATRICES 61

7.2 Weak to strong coupling via density matrices

This section considers the case in which a single cavity mode must be treated beyond a Markovian approximation, by considering the Jaynes- Cummings model, along with relaxation of both the two-level system and the cavity mode. The model is thus given by: d κ ρ = −i [H, ρ] − a†aρ − 2aρa† + ρa†a dt 2 Γ0 − σ+σ−ρ − 2σ−ρσ+ + ρσ+σ− (7.17) 2 g H = σ+a + σ−a† + σz + ω a†a. (7.18) 2 0 One should note that in this equation, a† is the creation operator for a pseudo-mode of the system, i.e. it does not describe a true eigenmode; the true eigenmodes are instead superpositions of modes inside and outside the cavity, just as in the previous section. Because the pseudo-mode overlaps with a range of true eigenmodes, the probability of remaining in the pseudo- mode will decay (see question 7.1 for the 1D case). This coupling between cavity modes and modes outside the cavity can be described in a Markovian approximation, leading to the decay rate κ in the above density matrix equation. In addition to the decay of the pseudo-mode, we also include a rate Γ0 describing decay of the two level system into modes other than the cavity mode. While in the one dimensional description considered previously, no such other channel exists, in three dimensions, if the cavity is not spherical, then decay into non-cavity directions is possible. In this case, one may consider Γ0 = ΓΩ/4π, depending on solid angle. For the experimental systems discussed below, this is generally the case, due to the need to have access to insert atoms into the cavity. More generally Γ0 describes the possibility of relaxation into any mode other than the cavity mode; in solid state contexts, other possible reservoirs often exist (e.g phonons, other quasiparticle excitations etc). Let us solve the equation of motion, Eq. (7.17), starting in an initially excited state | ↑, 0i. From this initial state, the only possible other subsequent states are | ↓, 1i and | ↓, 0i. As the last of these states cannot evolve into anything else it may be ignored, and we can write a closed set of equations for three elements of the density matrix:

PA = ρ↑0,↑0,PB = ρ↓1,↓1,CAB = ρ↑0,↓1 (7.19)

By taking appropriate matrix elements, one ﬁnds: d g P = −i (C∗ − C ) − Γ0P (7.20) dt A 2 AB AB A d g P = −i (C − C∗ ) − κP (7.21) dt B 2 AB AB B d g κ + Γ0 C = −i( − ω )C + i (P − P ) − C (7.22) dt AB 0 AB 2 A B 2 AB 62 LECTURE 7. CAVITY QUANTUM ELECTRODYNAMICS

2 One may further simplify these equations by noting that PAPB −|CAB| = 0 2 2 ∗ is conserved, and thus writing PA = |α| ,PB = |β| ,CAB = αβ . Substi- tuting this leads to the simpler equations:

d α −Γ0/2 −ig/2 α = (7.23) dt β −ig/2 i∆ − κ/2 β where we have written ∆ = ω0 − . In general this leads to two frequencies for decaying oscillations of the excitation probability:

Γ0 κ g2 iν − iν + i∆ − + = 0. (7.24) 2 2 4

Bad cavity limit — Purcell eﬀect in 3D In the limit of a bad cavity, i.e. κ g, Γ, the two frequencies corresponding to the above equation correspond separately to decay of the excited two- level system and decay of photons. Because these are on very diﬀerent timescales, one may treat the coupling perturbatively writing ν = −iΓ0/2+ δ, where at O(δ) the determinant equation becomes:

κ Γ g2 iδ i∆ − + + = 0, (7.25) 2 2 4 and since κ Γ this bad cavity limit gives:

i κg2 ∆g2 ν = − Γ0 + + . (7.26) 2 κ2 + 4∆2 κ2 + 4∆2

0 This describes a cavity-enhanced decay rate, Γeﬀ = Γ +Γcav, where the cavity decay rate describes the Purcell eﬀect, as discussed above. On resonance we have: g 2 1 Γ = 4 (7.27) cav 2 κ and recalling the earlier 3D results:

ω3|d |2 g 2 ω |d |2 0 ab 0 ab Γ = 3 , = (7.28) 3πε0c 2 2ε0V

(where we have assumed resonance, so ω0 = ), one may write:

3 3 Γcav 3πc 1 3 λ = P = 4 2 = 2 Q . (7.29) Γ 2V ω0 κ 4π V This closely resembles the one dimensional result before; the Purcell enhancement depends both on the quality factor, and also on the mode volume. An estimate of the minimum decay rate oﬀ resonance in this case 2 κ 3 1 λ3 yields Γ = Γ = 2 . However, if the back- cav,min cav,max 2ω0 16π Q V ground decay rate Γ0 also exists, this oﬀ-resonant reduction is not as relevant as the on-resonant enhancement. 7.3. EXAMPLES OF CAVITY QED SYSTEMS 63

Strong coupling — Rabi oscillations If the cavity is suﬃciently good, then rather than decay, the excitation probability oscillates. In the resonant case of ∆ = 0, one can write the general solution for the oscillation frequency as: s (Γ + κ) g2 Γ − κ2 ν = −i ± − . (7.30) 4 4 4

In order that Rabi oscillations exist, it is thus necessary that g > (Γ−κ)/2. In order that they should be visible before decay has signiﬁcantly reduced the amplitude, it is necessary that one has g > κ, Γ. In this limit, one has strong coupling, and decay is indeed very strongly modiﬁed.

7.3 Examples of Cavity QED systems

Having discussed the role of the parameters g, κ, Γ, and their relation to whether a system is weak or strong coupling, we now consider a variety of experimentally studied cavity QED systems, discussing the values of these parameters, and the advantages/disadvantages of the diﬀerent systems. A summary of the characteristic values is given in table 7.1; the discussion below highlights the origins of some of these parameters.

System Atom[1, 18] Atom[4] SC qubit[3] Exciton[5] Optical Microwave Microwave Optical κ/2π 1 MHz 1 kHz a 30 MHz 30 GHz Γ/2π 3 MHz 30 Hz 3 MHz b 0.1 GHz g/2π 10 MHz 50 kHz 100 MHz 100 GHz ω/2π 350 THz 50 GHz 10 GHz 400 THz λ3/V 10−5 10−1 λ/a = 1 10−1 Q 108 108 104 104 Other tmax tﬂight ∼ 100µs 1/T2 ∼ 3MHz 1/T2 ∼ 3GHz

aRecent work reaches 10Hz[19] bNot well known, dominated by cavity induced decay Table 7.1: Characteristic energy scales for diﬀerent cavity QED realisations, as discussed in the text.

Optical transitions of atoms We first consider real atoms, and cavities designed to be resonant for optical transitions of these atoms. In this case, the background atomic decay rate Γ0 is determined by the intrinsic properties of the atomic transition, reduced by the geometry of the cavity, and so this sets the scale that must be overcome for strong coupling. High quality mirrors at optical wavelengths can be made using dielectric Bragg mirrors. These consist of alternating layers of materials with different dielectric constant, with spatial period λ/4, so that even if the dielectric contrast between adjacent layers is not 64 LECTURE 7. CAVITY QUANTUM ELECTRODYNAMICS sufficient to cause strong reflection, the interference of multiple reflections will lead to strong overall reflection. To allow access for atoms to enter the cavity, the cavity length must be much larger than λ ' 0.8µm, and a typical size[1, 18] is a cavity length L ∼ 200µm, and beam waist w ∼ 20µm, which gives a ratio λ3/V ∼ 10−5. This small ratio means that a very high Q factor is needed to reach strong coupling, and Q ∼ 108 is possible in current experiments. Typical values of the corresponding g, Γ, κ are given in Table 7.1. In addition to the timescales included within Eq. (7.17), in some experimental systems, there is another timescale, that of the time for an atom to leave the cavity, however this is typically larger than all other timescales.

Microwave transitions of atoms Remaining with atoms as the “matter” part of the Jaynes-Cummings Hamil- tonian, some of the problematic features above can be removed if one considered instead microwave frequency transitions of the atoms. This increases the wavelength, allowing cavities comparable to the wavelength, and can also significantly improve the cavity quality. For microwave frequencies, superconducting cavities can be used (since the frequency is less than the BCS gap, ∆BCS ∼THz, so the superconducting mirrors are dia- magnetic at these frequencies. The remaining limitation on cavity quality instead comes from scattering off the mirror surface, and any gaps required in the cavity for atom injection. For niobium cavities operating at 50GHz (λ ' 6mm), values of λ3/V ' 0.1 and Q ∼ 1010 are possible[19]. The relevant atomic transitions at these frequencies are transitions between highly excited atomic states (Rydberg states). For states with n, l 1, the outermost electron remains far from the nucleus and so sees only the screened charge of +1, so the energy levels of such orbits are almost Hy- drogenic. For large n but l ' 0, the orbit is perturbed by the enhanced potential near the nucleus, which can be incorporated by a quantum defect, δl to write: Ry En,l ' − 2 , (7.31) (n − δl) with δl → 0 for large l. It is thus clear that transitions between n, n − 1 3 will have small energies, ∆En ∝ 1/n , and so for n ' 50 one can have microwave frequency transitions. Atoms can be prepared in such states by using tuned laser pulses. As well as allowing better cavities at these frequencies, Rydberg atoms also have significantly reduced spontaneous decay rates without sacrificing coupling to the cavity mode. The total decay rate from atomic state nl is given by: X 3 2 Γtot ∝ ωnl,n0l0 |dnl,n0l0 | . (7.32) n0l0 Selection rules imply that l0 = l ± 1, which means that two different limits of decay rates exist, depending on whether l = n−1 or l n, as illustrated in Fig. 7.3. 7.3. EXAMPLES OF CAVITY QED SYSTEMS 65

s p d g h (a) f

(b) 4 4 4 4 3 3 3

2 2 Figure 7.3: Cartoon of atomic level scheme, and transitions from a highly excited state with (a) l n, or (b) l = n − 1. Note that the actual values used are of the order n, l ∼ 50, much larger than shown here. l n In this case, decay to almost all other values n0 < n is possible (see arrows (a) in Fig. 7.3). We may consider the characteristic decay rate to nearby levels n0 ' n, or to the ground state n0 = 1.

• Decay to n0 ' n has ω ∝ 1/n3, and the dipole matrix element depends on the characteristic size of the orbit for large n, i.e. d ∝ n2 (this holds because both initial and ﬁnal states have −5 similar extensions). These combine to give Γn0'n ∝ n . • Alternatively, for transitions to the ground state, ω is n independent, being more or less ﬁxed at the the Rydberg energy. In this case the dipole matrix element is however much smaller, as it now involves the overlap between a large Rydberg state and a much smaller ground state orbital.√ Thus, the dipole matrix 3 −3 element depends on the overlap d ∝ 1/ n , giving Γn0'1 ∝ n .

As such, the latter process dominates for large n, and so the overall decay rate is a factor n−3 slower than for low excitation states. l ' n − 1 In this case, the only transition allowed by selection rules is to n0 = n − 1. Thus, this has the decay rate in the ﬁrst part of the above case, Γ ∝ n−5 and is yet even slower than for l n. These states with l = n − 1 are known as circular Rydberg states, as they correspond closely to classical circular atomic orbits, as expected from the correspondence principle.

While the decay rates are significantly reduced by the reduced energy of nearby transitions, and reduced overlap of remote transitions, the coupling 66 LECTURE 7. CAVITY QUANTUM ELECTRODYNAMICS to confined radiation for nearby transitions is not reduced in the same way. This is because the continuum density of states does not enter the √ calculation of g. Instead, one has g ∝ ωd for resonant transitions, giving √ g ∝ n, which increases with n. However, compared to√ optical transitions, the value of g is significantly reduced because g ∝ 1/ V , and the cavity has V ' λ3. Thus, the value of g in table 7.1 is reduced compared to optical transitions, but is reduced by a smaller factor than the reduction of the atomic decay rate.

Superconducting qubits in microwave resonators Continuing with microwave frequencies, one can also consider “artiﬁcial atoms” coupled to microwave cavities; these have the advantage of considerably increasing the coupling strength, by (in this case) increasing the number of electrons involved in the artiﬁcial atom.

Waveguide Cg Cin

Lr CCr Q EJJE Superconductor V

Figure 7.4: Schematic diagram of superconducting qubit capacitively coupled to a stripline microwave resonator (left), and the equivalent circuit (right), coloured for comparison of diagrams — after Koch et al. [20].

Figure 7.4 illustrates one schematic design of a superconducting island capacitively coupled to a stripline resonator. This particular design is known as the “transmon” qubit[20], as the atomic states corresponds to quantised modes plasmon oscillations, modified by the transmission line. The equivalent circuit shows that the artificial atom consists of a pair of Josephson junctions, shunted by a large capacitance. The pair of junctions exists so that one may tune the effective Josephson coupling via a magnetic flux, giving EJ,eff = EJ cos(ΦB/Φ0), giving:

2 2 e (nQ − nbias) Hatom = − EJ,eff cos(φQ), (7.33) 2CQ,eff where the number and phase operators obey canonical commutation relations. This qubit is capacitively coupled to the resonator mode, i.e. the charge difference across the resonator conductors voltage biases the qubit. This coupling can be engineered to be relatively large, of the order of 100MHz. In other designs of qubit, the coupling is reported to be able to be made yet larger, such that g ≥ ω. This limit is referred to as ultra-strong coupling, in which the rotating wave approximation is invalid. Unlike the microwave system with Rydberg atoms, where atoms eventually leave the cavity and can be measured, the observation of these superconducting systems is typically via the emitted radiation. For this reason 7.4. FURTHER READING 67

κ is chosen to be larger than physical constraints would require, so that sufficient photons escape to allow measurement. The origin of the artificial atom decay rate in these experiments is not particularly clear, since coupling to bulk radiation modes should be negligible. However, since the system is in a solid state environment, other degrees of freedom exist that can lead to relaxation. These other degrees of freedom certainly lead to dephasing, so that in these systems, 1/T2 is a significant rate, arising from charge and flux noise on the superconducting circuit.

Quantum-dot excitons in semiconductor microstructures

Finally, we consider artificial atoms at optical frequencies. One example of this concerns excitons in quantum dots, coupled to semiconductor microstructures. The exciton states can be regarded as Hydrogen-like wavefunctions of electrons and holes, with a reduced binding energy, and increased Bohr radius due to the dielectric screening εrel ' 10 and reduced electronic mass. Although the excitonic binding energy is much less than optical frequency, the relevant transition is the creation of a bound exciton, which corresponds to the band gap less the binding energy. Typical semiconductor band gaps are of the order of 1eV , leading to optical frequencies. Compared to real atoms, the notable improvement in table 7.1 is the significantly enhanced g; this arises from the increased Bohr radius, and hence larger dipole matrix element, as well as the much reduced mode volume. As the quantum dots are fixed inside the semiconductor, microstructures can be grown with characteristic sizes comparable to the wavelength of light. The mechanism of light confinement used for such cavities varies, combining one or more of dielectric contrast, Bragg mirrors, or photonic band gap materials. Dielectric contrast can produce reasonable confinement for whispering gallery modes (WGM) in circular resonators; in this case the WGM makes a shallow angle of incidence with the edge of the cavity, and so is confined by total internal reflection. Cavity decay rates are however typically much larger than cavities used for atoms, due to the lower quality of mirrors in integrated semiconductor heterostructures. The exciton decay rate is much larger than for real atoms due to the solid state environment. Just as for superconducting artificial atoms, this again leads both to reduced T1 and also T2 dephasing.

7.4 Further reading

For a discussion of the Purcell eﬀect, and the crossover between weak and strong coupling, see e.g. Meystre and Sargent III [12]. The discussion of particular cavity QED systems in this chapter is based on various reviews: for Rydberg atoms, see Raimond et al. [4] or Gallagher [21]; for superconducting qubits, see Blais et al. [3]; for excitons in semiconductor microcavities, see Khitrova et al. [5]. 68 LECTURE 7. CAVITY QUANTUM ELECTRODYNAMICS

Questions

Question 7.1: Pseudo-mode decay rate. Consider a standing wave conﬁned within the cavity, i.e. ψ0(x) = p2/a cos[(2n + 1)πx/a]. By considering the decomposition of this wave onto the true eigenmodes of the extended system (as calculated in Sec. 7.1), show that the time evolution of the overlap |hψ0|ψ0(t)i| decays at rate κ/2, as deﬁned in Eq. (7.14). Lecture 8

Collective eﬀects of two-level atoms: open systems, superradiance

So far in these lectures we have either considered behaviour of a single two-level atom, or if we have considered ensembles then we have assumed the atoms act independently. However, as first pointed out by Dicke[22], this approach cannot be correct if the atoms are close together, as they see the same electromagnetic modes with sufficiently similar phases, and so emission is a collective process. This collective behaviour can significantly change how decay occurs, even when the electromagnetic modes are treated as a Markovian bath, with short memory times. In this lecture we will consider the case of many atoms coupled to a continuum of radiation modes (i.e. without a cavity); the next lecture will consider collective effects of many atoms with a cavity.

8.1 Simple density matrix equation for collective emission

We may begin by considering coupling between a collection of atoms and radiation modes:

X X gk H = σ−a† ei(ωk−)t−ik·ri + σ+a ei(−ωk)t+ik·ri . (8.1) SR 2 i k i k k i

Repeating the derivation of the system density matrix equation, we recover:

Z t d X X gk 2 ρ(t) = − dt0 eik·(ri−rj ) dt 2 ij k h + − ∗ − + ∗ + − i × σi σj ξkρ − σj ρσi (ξk + ξk) + ρσi σj ξk . (8.2)

Two extremes of this equation exist; if |ri − rj| λ0, then the atoms will act independently. In the other extreme, we may neglect all phase factors,

69 70 LECTURE 8. SUPERRADIANCE in which case we can introduce: X J = σi (8.3) i and write: d Γ ρ(t) = − J +J −ρ − J −ρJ + + ρJ +J − . (8.4) dt 2 This second limit is what we will consider in this section. We must however note that when separations are small, one must also consider the effects of Coulomb interactions between the two-level systems. In the Coulomb gauge, this means the explicit Coulomb term will have an effect; in the dipole gauge, the problem is instead that long-wavelength modes (for which phase factors can not be neglected) describe the effects of Coulomb interaction, leading to important energy shifts. We will discuss this effect in section 8.2. Putting aside these issues for the moment, let us study the consequences of Eq. (8.4)

Dicke Enhancement of emission rate A simple understanding of how collective emission diﬀers from independent emission can be found by considering eigenstates of |J|,J z. These are spin states and so obey:

We may then use these states as a basis for the density matrix equation, and writing equations for the diagonal elements PMJ = ρMJ,MJ we ﬁnd:

d P = − Γ[J(J + 1) − M(M − 1)]P dt M,J M,J + Γ[J(J + 1) − M(M + 1)]PM+1,J (8.8)

The rate of emission is thus I = Γ[J(J + 1) − M(M − 1)] = Γ(J + M)(J − M + 1), whereas if incoherent, the total emission rate would be ΓN. If we consider cases where J = N/2, if J takes the maximum value possible consistent with the number of two-level systems, then the radiation rates at characteristic values of M are:

N M = 2 I = ΓN N 2 M = 0 I = Γ 4 (8.9) N M = − 2 + 1 I = ΓN Thus, near M = 0, there is macroscopic enhancement of the emission rate compared to the incoherent case. Even when M = 1 − N/2, the emission is much greater than it would be for a single excitation if considering independent atoms. 8.1. SIMPLE DENSITY MATRIX EQUATION FOR COLLECTIVE EMISSION 71 Solving the simple model The above estimates indicate the general idea of superradiance; as M decreases, the collective eﬀects increase in importance, leading to an increasing rate of transitions through the super-radiant cascade. Our aim in the following will be to describe this cascade. Before discussing a tractable approximation scheme, we may note that the Laplace transformed equations can be straightforwardly solved to give:

[s + Γ2J]PJ,J (s) = 1

[s + Γ(J + M)(J − M + 1)]PM,J (s) = Γ(J + M + 1)(J − M)PM+1,J (s)

The general result will thus be a sum of exponential decays with diﬀerent time constants, and prefactors linear in t (degenerate decay rates exist, with M ↔ 1−M having the same rate, hence the matrix problem for eigenvalues is defective). While this approach may be appropriate numerically, it does not help to produce a qualitative understanding of the decay, so we will instead concentrate on approximately solving the density matrix equations.

Semiclassical evolution after initial times As a ﬁrst approach, we may write a semiclassical approximation for how M evolves. Since J does not change during the time evolution, we will drop the label J, and assume J = N/2 throughout.

d X d X hMi = M P = −Γ [M − (M − 1)](J + M)(J − M + 1)P dt dt M M M M = −Γh(J + M)(J − M − 1)i (8.10)

The semiclassical approximation is to assume that h(M −hMi)2i hMi2 so that expectations of products of operators can be represented as products of expectations, and one has:

d hMi ' −Γ(J 2 − hMi2) (8.11) dt We have also assumed here that J, hMi 1, such an approximation will be seen later to arise naturally whenever the semiclassical approximation is valid. These equations can be solved by substituting hMi = J tanh(χ), which lead to dχ/dt = −ΓJ, hence the general solution is:

dhMi hMi = −J tanh[ΓJ(t − t )],I = −Γ = ΓJ 2sech2[ΓJ(t − t )] D dt D (8.12)

Semiclassical results in Heisenberg picture The same results can be recovered in the Heisenberg picture by considering equations of motion for the operators J z,J ±. This approach will be useful later on, when considering an extended system. Using the commutation relations [J +,J −] = 72 LECTURE 8. SUPERRADIANCE

J ΓJ

0 -d / dt

-J 0 tD tD Figure 8.1: Time evolution of hMi and the associated rate of radiation calculated semiclassically

2J z, [J z,J ±] = ±J ±, we may write: d dρ Γ hJ −i = Tr J − = − Tr [J −J + − J +J −]J −ρ = ΓhJ zJ +i. dt dt 2 (8.13) Similarly, one ﬁnds (d/dt)hJ zi = −ΓhJ +J −i. The semiclassical approximation corresponds again to factorising products of operators, thus yielding: d d hJ zi = −Γ|hJ −i|2, hJ −i = ΓhJ zihJ −i (8.14) dt dt Clearly hJ zi = hMi, and so the previous equations can be recovered by writing hJ zi = J tanh(χ) along with hJ −i = Jsech(χ)eiφ. The phase φ is constant, and is not determined by the equations of motion. While this substitution is most suitable to solve the equations, it is also worth noting (for future reference) that another substitution is more natural in order to understand the behaviour. By noting that the components J −,J z represent parts of a collective spin, one is led to write:

hJ zi = J cos(θ), hJ −i = J sin(θ)eiφ, → θ˙ = ΓJ sin(θ). (8.15)

Hence, superradiance corresponds to the collective spin behaving as a damped pendulum, initially in its inverted state.

Early time evolution In the above description, it is clear that at early times, when fully inverted, the classical equations will not start. However, at early times the semiclassical approximation fails. In order that semiclassics is valid, the distribution of M should not spread too much, and so the M dependence of the time evolution rate should be small compared to the mean evolution, i.e.

d 2 2 (J + M)(J − M + 1) = |2M − 1| J − M . (8.16) dM It is clear that this is true as long as |M| J, thus semiclassical evolution describes the late time evolution, but not the early time. At early times, 8.1. SIMPLE DENSITY MATRIX EQUATION FOR COLLECTIVE EMISSION 73

M spreads, but at later times no significant spread occurs. This allows us to write: Z P (t) = dsP (t )δ M − hM(t)i| . (8.17) M J−s 0 hM(t0)i=J−s i.e., if we find the density matrix at early times, then for each possible value of M arising from it, we may evolve forward semiclassically. The initial state, can be matched to the solution in Eq. (8.12), expanded for small s to give: h i 2ΓJ(t0−tD) hM(t0)i = J − s ' J 1 − 2e (8.18) thus the “delay time” appearing in the semiclassical equation is given by: 1 2J t (s) = t + ln . (8.19) D 0 2ΓJ s This formula allows one to translate the result of the early time evolution at t0 into the subsequent evolution of M. Small values of s, describing small deviations from an initially inverted state, correspond to longer delay times; because the inverted state is unstable the initial motion away from this point is exponential growth, hence the logarithmic dependence of delay time on the value of s. To complete the problem, we must find how the distribution of M spreads for early times. At early times, excitation numbers remain of order M ' J, i.e. we may write PJ=M−s and expand Eq. (8.8) for small s to give: d P = −Γ(2J − s)(s + 1)P + Γ(2J − s + 1)sP dt s s s−1 ' −Γ2J[(s + 1)Ps − sPs−1]. (8.20)

Solving the ﬁrst two cases, with the initial condition P0 = 1,Ps>0 = 0, one ﬁnds: −Γ2Jt −Γ2Jt −Γ2Jt P0 = e ,P1 = e (1 − e ),... (8.21) from which one may guess the general solution:

−Γ2Jt −Γ2Jt s Ps = e (1 − e ) (8.22) This can be shown to satisfy the equation of motion, i.e.: d P = −Γ2J P + s(e−Γ2Jt − 1 + 1)P ) dt s s s−1 = −Γ2J [(1 + s)Ps − sPs−1] . (8.23)

s This is a Bose-Einstein distribution, Ps ∝ z with z = 1−exp(−Γ2Jt) which has mean excitation hsi = z/(1 − z) = eΓ2Jt − 1, which grows exponentially at early times. If we choose t0 such that 2ΓJt0 1, then (for large enough J) it is possible simultaneously to fulﬁl the condition for semiclassical evolution, and also to expand the above distribution for large t as giving:

−Γ2Jt0 Ps(t0) = exp −Γ2Jt0 − se (8.24) 74 LECTURE 8. SUPERRADIANCE

Putting together all of the above ingredients, one has the result: Z −Γ2Jt0 PM (t) = ds exp −Γ2Jt0 − se 1 2J × δ M + J tanh ΓJ t − t − ln (8.25) 0 2ΓJ s

Since t0 was somewhat arbitrary, this result should not depend on t0, and indeed (see question 8.2) it does not. Thus, one can set t0 = 0 and ﬁnd a full semiclassical approximation of the time evolution of PM .

Early time evolution for Heisenberg picture The above analysis of PM for short times has an equivalent manifestation in the Heisenberg picture. For early times, one cannot factorise hJ +J −i as |hJ −i|2, but instead, one needs to worry about correlations of the operators. This can be treated approximately by assuming that the initial value of hJ −i should be drawn from a Gaussian distribution of mean 0, and variance hJJ|J +J −|JJi = 2J. These quantum fluctuations are then amplified by the semiclassical behaviour, just as the semiclassical evolution of PJ amplifies small initial differences in M via the different classical trajectories they lead to.

8.2 Beyond the simple model

The above discussion assumed a contradictory set of conditions; it assumed that the atoms are close enough that phase diﬀerences between their coupling to light could be neglected, and that also assumed that the Coulomb interactions between the atoms could be neglected. As the following will show, one or other of these conditions can hold, but not both, and their violation will modify the superradiant behaviour.

Coulomb interactions and dephasing If the atoms remain close together, the above treatment of coupling to propagating radiation modes is correct, but Coulomb interactions between atoms will break the indistinguishability that led to the collective enhance- ments. Indistinguishability is broken as each atom sees a diﬀerent environment of Coulomb interactions from its neighbours. The Coulomb interaction can be written as:

X + − + − Hcoulomb = Ωij σi σj + σj σi . (8.26) i>j

Treating atoms in the dipole approximation, the dipole-dipole coupling is given by: 2 ˆ ! |dab| (d · rij) Ωij = 3 1 − 3 2 . (8.27) 4πε0rij rij Before discussing the eﬀect of this term, we may ﬁrst notice that it depends on rather similar parameters to Γ, and in terms of the characteristic 8.2. BEYOND THE SIMPLE MODEL 75 strength, one may write:

2 3 3 Ωij |dab| 3πε0c 3 λ ' 3 2 3 = 3 (8.28) Γ 4πε0rij |dab| ω0 32π rij

Thus, the same condition required in order to neglect phase diﬀerences also implies that Ωij & Γ. Superradiance occurred because of constructive interference between the indistinguishable pathways of atomic de-excitation. The Coulomb term makes atoms distinguishable. Physically this means that while the coupling to light in Eq. (8.4) causes transitions between symmetric atomic states, Coulomb interaction make symmetric superpositions evolve into less symmetric superpositions, thus reducing J. Coulomb interactions do not however change the excitation level M, it merely transfers excitation between atoms. As such, one may estimate the relative size of eﬀect of the Coulomb term vs M by considering the number of possible states with a given M. When M = N/2, only one state is possible, the symmetric state with J = M = N/2, thus Coulomb interactions may shift the total energy, but not modify the state. For smaller values of M, one may use the number of representations of N spins with modulus J, given by:

(2J + 1)N! ν (J) = . (8.29) N (N/2 + J + 1)!(N/2 − J)!

The number of possible states of given M is the number of representations PN/2 having J > M, i.e. J=M νN (J). This behaviour leads to the following scenario. For M ' J, Coulomb terms do not√ significantly effect the decay, but as M approaches J (in fact, when M ' J), Coulomb terms√ rapidly dephase the atoms, transferring the system to states with J ≤ N, and thus suppressing superradiance. If one were to work in the electric Dipole gauge, the equations may appear different, as no static Coulomb term exists in the Hamiltonian. However, in this case, the low k part of the photon mediated interaction describes the same physics. This means that even though a Markovian approximation can be made for the sum in Eq. (8.2) for nearly resonant modes, the low k modes must be treated carefully, and contribute an important imaginary part (energy shift). In contrast, the different form of gk in the Coulomb gauge means that the low k parts of this sum cause no particular problem, as the Coulomb term is already explicitly included.

Extended systems To avoid the effects of Coulomb interactions, one may instead consider extended systems, where rij > λ, and so Coulomb effects become weaker. However, in this case, we can no longer neglect the phase differences seen by different atoms, and so the evolution will be more complicated that Eq. (8.4). 76 LECTURE 8. SUPERRADIANCE

The simplest possible extended system is a one dimensional system1. In this case, we may replace the phase factors by exp[ik(zi − zj)], and forget about summation over directions of k. Note that in this replacement, we have chosen to speciﬁcally consider forward propagating waves in the z direction. An equivalent description of backward propagating waves will exist, however for simplicity we neglect this complication. In this approximation, our equation of motion becomes:

Z t d X X gk 2 ρ(t) = − dt0 eik(zi−zj ) dt 2 ij k h + − ∗ − + ∗ + − i × σi σj ξkρ − σj ρσi (ξk + ξk) + ρσi σj ξk . (8.30) and we may note that:

Z t 0 ∗ 1 1 dt ξk = + = −iP + πδ(ωk − ) (8.31) i(ωk − ) + 0 ωk − In the previous discussion of decay, we have only considered the delta function, and have neglected the principal value part. This can sometimes be valid, as the imaginary part gives an energy shift — i.e. it contributes a term that looks like [Heff, ρ], describing Hamiltonian dynamics. If this Lamb shift is site independent (or if we only have a single atom) then such a shift can be absorbed into renormalisation of the bare Hamiltonian. How- ever, it can also lead to effective interactions between the atoms. We have already mentioned one example of this; Coulomb interactions in the Dipole gauge originate from the low k contribution of these imaginary parts. In the current case, we work in the Coulomb gauge, so these low k terms are not important, however the imaginary parts of the resonant terms (i.e. for ωk ' ) will play an important role in the extended system. These terms lead to phase shifts that are responsible for causality emerging in the final result. To see this causality, we consider:

g 2 eik(zi−zj ) X k Λij = + . (8.32) 2 i(ωk − ) + 0 k Concentrating on nearly resonant terms, we may approximate the density of states by Γ/2π, and extend the sum over k to ±∞, yielding:

Γ Z ∞ eik(zi−zj ) ik0(zi−zj ) Λij ' cdk + = ΓΘ(zi − zj)e (8.33) 2π −∞ i(ck − ) + 0 where k0 = /c. In the case that we may neglect phases, this becomes Γ/2, due to the step function. If we phase shift the system operators

1To be able to truly treat a 1D system, one has two opposing constraints; the system must be narrow enough that there are not significant numbers of transverse modes. At the same time, diffraction must be small enough that√ propagation remains axial. These constraints are best satisfied by a tube of waist w ' Lλ. 8.2. BEYOND THE SIMPLE MODEL 77

− − −ik0zi σi → σi e then we ﬁnd:

dρ X = −Γ σ+σ−ρ − σ−ρσ+ − σ−ρσ+ + ρσ+σ− (8.34) dt i j j i i j j i zi>zj

This describes causality, in that operators must always proceed from small z to larger z. Note that the previous factor of 1/2 is replaced by the restricted range of the sum. Because we have written a Markovian density matrix equation, we have neglected propagation time of electric ﬁelds, and so the above equation only applies for intermediate sized systems. For larger systems Heisenberg equations can be used to extend the range of validity. To describe the behaviour more concretely, let us consider the Heisen- berg equations within this Markovian approximation. Following the procedure of Sec. 8.1 we may write:

d X n o hσzi = −Γ Tr [σz, σ+]σ− − σ+[σz, σ−] ρ dt k k i j j k i zi>zj X + − + − = −Γ hσk σj + σj σk i (8.35) zj

These equations cannot directly be factorised if they apply to single two-level systems, but a coarse grained version, in terms of:

X z X − N (z) = δ(z − zi)hσi i, P(z) = δ(z − zi)hσi i, (8.37) i i can be considered instead. These give the semiclassical approximation

dN Z z = −Γ dz0 P(z)P∗(z0) + P(z0)P∗(z) (8.38) dt dP Z z = 2Γ dz0N (z)P(z0) (8.39) dt

If we deﬁne E = R z dz0P(z0) then this reduces to the simpler set of equations: dN dP dE = −Γ[PE∗ + EP∗] , = 2ΓNE, = P. (8.40) dt dt dz Here, E can be interpreted as proportional to the electric ﬁeld strength, resulting from the integrated polarisation of the medium to the left of the current position. (Were we to have instead written Heisenberg equations from the begining, and were we to have avoided making the Markovian approximation, then a similar set of equations would have been found, but with the retarded time τ = t − z/c appearing in place of the t.) 78 LECTURE 8. SUPERRADIANCE

Equations (8.40) can then be understood by the same substitution as in Sec. 8.1, mapping them to a generaised damped pendulum problem. iφ Writing N = N0 cos(θ), P = N0 sin(θ)e one has the equations: dθ dE eiφ = 2ΓE, = N sin(θ)eiφ (8.41) dt dz 0 giving the result:

2 2 d θ 2 dθ = 2ΓN0 sin(θ),I ∝ |E| ∝ . (8.42) dzdt dt Compared to the small system equation, the principal difference here is that this equation is now second order, i.e. it contains both damping and inertia. For sufficiently long samples, one thus has ringing oscillations, as each atom accumulates a different phase depending on where it is. This causes overshoot, and hence oscillations.

8.3 Further reading

A comprehensive review of superradiance may be found in Gross and Haroche [23].

Questions

Question 8.1: Coherent states of spin Consider an alternative choice of coherent atomic states, known as coherent states of spin:

|ψ(α)i = N eαJ+ |−J, Ji . (8.43)

8.1.(a) Show that normalisation of |ψ(α)i implies N (1 + |α|2)J = 1.

8.1.(b) Evaluate hψ(α)| Jz |ψ(α)i and hψ(α)| J+J− |ψ(α)i.

8.1.(c) Thus, compare the rate of photon emission from a coherent state of spin and the Dicke states discussed earlier in this section.

Question 8.2: t0 independence of superradiance Ps(t) Show that equation (8.25) is independent of t0, by evaluating the partial derivative with respect to t0 and showing it vanishes. Lecture 9

More collective eﬀects: Cavity system and the Dicke model

In lecture 8, collective atomic decay was discussed for coupling to a continuum of modes. This lecture considers the case of collective atomic behaviour in a cavity. Two particular cases are considered: the first section is on the possibility of spontaneous coherence in thermal equilibrium, and the second section discusses the general time evolution starting from a fully excited atomic state. In the first section we ask whether the model in thermal equilibrium can undergo a phase transition in which the two-level systems polarise, and generate an expectation of the photon field.

9.1 Phase transitions, spontaneous superradiance

In this first section, we will consider the Dicke model without the rotating wave approximation. Written in terms of spin operators σi for the two-level systems, this is: X X g H = σz + ωa†a + a† + a σ+ + σ− . (9.1) i 2 i i i i We wish to consider the thermodynamics of this model, by considering a mean-field theory, in which we assume a coherent mean-field for the photon state, and find the associated free energy. If the free energy is minimised by a non-zero photon field, (i.e. if such a state has a lower free energy than the vacuum state), then there is a phase transition to a spontaneous superradiant phase. For a coherent photon state exp(−|α|2/2+αa†)|0i, the associated partition function can be written as:

−βω|α|2 Y X 0 x z Z[α] = e exp −β 2gα σi + σi . (9.2) i σi Here, we have written real and imaginary parts of α as α = α0 + iα00, and + − x have used σi + σi = 2σi . One may then evaluate the sum over states of

79 80 LECTURE 9. THE DICKE MODEL the two-level system by transforming to the diagonal basis in the presence of ﬁeld α0. The eigenvalues of the 2 × 2 matrices are then ±E/2, with E = p2 + 4g2α02, and so the free energy is given by:

N β p F (α) = ω|α|2 − ln cosh 2 + 4g2α02 . (9.3) β 2

To determine where a non-zero value of α can minimise F , we should ﬁnd the derivative of F with respect to α, to locate possible stationary points. Clearly, dF/dα00 = 2ωα00, and so for any stationary point we must have α00 = 0. For the real part of α we have instead

dF N β 4g2α0 β p = 2ωα0 − tanh 2 + 4g2α02 . (9.4) dα0 β 2 p2 + 4g2α02 2

Using the eigenvalue energies E deﬁned above, this may be rewritten in the simpler form: dF Ng2 βE = 2α0 ω − tanh . (9.5) dα0 E 2 A non-zero α0 solution exists if the term in brackets can be made to vanish. In order that this vanishes at some temperature T > 0, we require that the function tanh(βE/2)/E crosses the value ω/Ng2. As illustrated in Fig. 9.1, at small temperatures, tanh(βE/2)/E ' 1/E, while at large temperatures it becomes 1/2T . Thus, for a crossing to occur, one requires 1/E > ω/Ng2, and since E > ||, a transition may exist if < Ng2/ω.

1 Small T: tanh(βE/2)/E = 1/E

0.8

0.6 E/2)/E 2 β ω/N g 0.4 tanh( Large T: Decreasing E tanh(βE/2)/E = 1/2T 0.2

0 0 1 2 3 4 5 Temperature

Figure 9.1: Graphical representation of Eq. (9.5). It is clear that for a solution to exist, it is necessary for 1/|| > 1/E > ω/Ng2.

Now, recalling the deﬁnition of g in terms of microscopic parameters and matrix elements in Eq. (1.39), we have the condition:

N 2|d |22 ω < ab . (9.6) V ε0ω 9.2. NO-GO THEOREM: NO VACUUM INSTABILITY 81

Thus, apparently a transition can occur if the density of two-level systems is large enough. This phase transition was originally discussed by Hepp and Lieb [24, 25], and in the form presented here by Wang and Hioe [26].

9.2 No-go theorem: no vacuum instability

If the density of two-level systems is large enough, the Dicke model is invalid, as we have neglected the matter-radiation coupling originating from the A2 terms. Let us restore these terms, and see at what density of two- level systems they become important. In the same single-mode approximation in which we are treating the quantised radiation, the correction due these terms can be written following Eq. (1.35) as:

2 † 2 q δHA2 = Nζ(a + a ) ; ζ = . (9.7) 4mε0ωV

(NB, the mass m appearing here for the contribution from a single atom is 1/m = 1/me + 1/mi, i.e. is the reduce electron-ion mass.) Including this term in the Hamiltonian, the function F (α) acquires an extra dependence on α0, giving

N β p F (α) = ω|α|2 + ζN4α02 − ln cosh 2 + 4g2α02 . (9.8) β 2

In terms of the condition for a phase transition to occur, this means one should replace ω → ω + 4ζN; i.e. the condition is now (ω + 4ζN) < Ng2 Physically, this means that the A2 terms in the Hamiltonian describe a dielectric response of the atoms, opposing large transverse fields; as a result, the energy cost of spontaneous polarisation has increased, and sponatneous effects can arise only if the two-level system susceptibility is larger than it was above. Combining the defintions of ζ and g, we may write the new condition as: N q2 N 2|d |22 ω + < ab . (9.9) V mε0ω V ε0ω As a function of density of two-level systems, this condition is clearly not satisfied at N/V = 0, but will be satisfied for some finite density of two-level systems if and only if: q2 2|d |22 < ab . (9.10) mε0ω ε0ω Thus, spontaneous polarisation requires that the energy and dipole matrix 2 2 elements of an atom can satisfy 2|dab| > q /m.

Thomas-Reiche-Kuhn sum rule The energy and dipole matrix elements obey a sum rule, restricting the relative values they may take. This sum rule can be derived from assuming that the internal atomic Hamiltonian depends on internal momentum like 82 LECTURE 9. THE DICKE MODEL

2 H0 ' p /2m + V (r). (Such an assumption was already implicit in our derivation of g.) Then, as before, p/m = i[H0, r], and so 1 1 r, [H , r] = −i [r, p] = . (9.11) 0 m m Since this identity gives a c-number, the expectation of this identity for any state will match, thus:

q2 = hψ| qr, [H , qr] |ψi . (9.12) m 0 Then, inserting a complete set of states |φi, we have:

q2 Xn o = hψ| qr |φi hφ| [H , qr] |ψi − hψ| [H , qr] |φi hφ| qr |ψi m 0 0 φ X = {dψφ(Eφ − Eψ)dφψ − dφψ(Eψ − Eφ)dψφ} φ X 2 = 2 |dψφ| (Eφ − Eψ). (9.13) φ

This is the Thomas-Reiche-Kuhn sum rule, relating the total oscillator strength, weighted by energy, to the charge and mass of particles involved. There is an immediate corollary of this rule; suppose that ψ is chosen to be the ground state. In this case, the right hand side of Eq. (9.13) is a sum of positive terms, and so is greater than any one of its terms. Taking the two lowest levels, one may then write:

q2 > 2|d |2. (9.14) m ab The sum rule has therefore proved the opposite inequality to the one we require for a phase transition; the sum rule prevents such a phase transition from occurring. This observation was ﬁrst pointed out by Rz¸a˙zewski, W´odkiewicz,and Zakowicz˙ [27].

9.3 Radiation in a box; restoring the phase transition

The above result shows that the phase transition by which the vacuum state becomes unstable, and spontaneously polarises is an artefact of that model, and not physical. This however is not the end of the story. There is a simple extension to the Dicke model, which is appropriate in a variety of recent works on confined quantum optical systems, which can restore the phase transition. That extension is to consider a closed system with a density of excitations. This means considering the at-first counterintuitive idea of a chemical potential for photons. The idea of a chemical potential for photons is not generally considered because photons cannot easily be confined, and so a fixed density of photons 9.4. DYNAMIC SUPERRADIANCE 83 is hard to achieve. Without confinement, photons can be exchanged with the bulk, which acts as a reservoir at zero chemical potential. However, in engineered cavity quantum electrodynamics systems, photon confinement is exactly what is being created, and so for such systems, including polaritons in semiconductor microcavities, Josephson junctions in microwave waveguides, and atoms in superconducting mirror cavities, a chemical potential for photons is a useful concept. Adding a chemical potential, we have

X 1 H → H˜ = H − µM, M = σz + + a†a. (9.15) i 2 i

The net result is to replace → ˜ = − µ and ω → ω˜ = ω − µ in the previous Hamiltonian (although the factors of and ω in the coupling strengths remain unmodiﬁed). In this case, it is clear that one can satisfy the condition ˜(˜ω + 4ζN) < Ng2 by ensuring the chemical potential is close enough to the two-level system energy. Since this may be done at any density of two-level systems, we may assume a low density, and so neglect ζ — i.e. neglect the A2 terms in the Hamiltonian. Since a transition may occur, it is interesting to ﬁnd the critical temperature of this transition. From the adapted form of Eq. (9.5), this condition is: − µ ( − µ)(ω − µ) = Ng2 tanh . (9.16) 2T

This can be combined with the equilibrium expectation of the number of excitations hMi in the absence of an expectation of α, which gives.

hMi 1 − µ = 1 − tanh . (9.17) N 2 2T

Combining these allows one to plot the phase boundary, as shown in Fig. 9.2. See question 9.1 for further discussion.

9.4 Dynamic superradiance

In this section, we consider the general time evolution of the Dicke model, starting from a fully inverted atomic state. This describes the analog of the superradiance discussed in Sec. 8 in the case where there is a good cavity. For illustration, we discuss here the simplest case of = ω, and so our model is: g H = ωJ z + a†J − + aJ + + ωa†a. (9.18) 2

To study the semiclassical dynamics, we ﬁrst write the Heisenberg equa- 84 LECTURE 9. THE DICKE MODEL

5 5 4

4 /T N __ N

__ 3 √ √ g 3 2

1 -6 -5 -4 -3 -2 10 10 10 10 10 2 __ ω/g√N =0.0

Temperature: T/g __ 1 ω √ /g __N =0.5 ω/g√N =1.0 0 0 0.1 0.2 0.3 0.4 0.5 Density: 〈M〉/N

Figure 9.2: Critical temperature vs excitation density, plotted for three diﬀerent values of ω (measured with respect to ), including the case of ω = discussed in Q. 9.1. The inset shows 1/T vs density on a logarithmic scale, illustrating the asymptotic form at low densities. tions of motion: g J˙x = i[H,J ] = −ωJ y + i a − a† J z, (9.19) x 2 g J˙y = i[H,J ] = +ωJ x − a + a† J z, (9.20) y 2 g g J˙z = i[H,J ] = − i a − a† J x + a + a† J y, (9.21) z 2 2 g ia˙ = −[H, a] = (J x − iJ y) + ωa. (9.22) 2 The semiclassical approximation is then to replace these equations for non- commuting operators for equations for their commuting expectations, and to factorise the expectations, so h(a + a†)J zi → ha + a†ihJ zi. Doing this, and writing hai = α = α0 + iα00 the above equations can be written as:

 gα0  g J˙ = h × J, h = −gα00 , iα˙ = (J − iJ ) + ωα. (9.23)   2 x y ω

In order to solve these equations, it is ﬁrst convenient to transform to a rotating frame. This means substituting α → αe−iωt, and J − → J −e−iωt, i.e.  cos(ωt) − sin(ωt) 0  J →  sin(ωt) cos(ωt) 0  J. (9.24) 0 0 1 This then gives  gα0  g J˙ = h × J, h = −gα00 , iα˙ = (J − iJ ) . (9.25)   2 x y 0 9.4. DYNAMIC SUPERRADIANCE 85

At this point, starting in the excited state J = J0zˆ, we can make the ansatz:

 0   0  ˙ ˙ J = J0  sin(θ)  , J = J0θ  cos(θ)  . (9.26) cos(θ) − sin(θ)

Then, substituting this into the equations of motion, we have:

 0   −α00 cos(θ)  ˙ 0 J0θ  cos(θ)  = gJ0  −α cos(θ)  , (9.27) − sin(θ) α0 sin(θ) which implies α00 = 0 and θ˙ = −gα0. Then, the equation for α = α0 gives:

gJ α˙0 = − 0 sin(θ). (9.28) 2 Thus, the angle deﬁning the Bloch vector obeys the equation of motion:

g2J θ¨ = −gα˙0 = 0 sin(θ). (9.29) 2 This is the equation of motion for an (initially) inverted pendulum, and so the Bloch vector makes traces out great circles passing through the entirely inverted, and entirely empty states. One should compare this second order equation for dynamical superradiance in a cavity to the ﬁrst order equation in the decaying case discussed in Sec. 8.1 of the previous lecture. When non-resonant, i.e. 6= ω, the comparable equations are more complicated (see question 9.2), however the general result is similar: after transformation to a rotating frame, the Bloch vector traces out circles (see Fig. 9.3), and can always be mapped to the problem of an inverted pendulum.

Figure 9.3: Bloch sphere, and paths that Bloch vector takes in the rotating frame, in the general (non-resonant) case. 86 LECTURE 9. THE DICKE MODEL

9.5 Further Reading

The original phase transition of the Dicke model is discussed in Refs.[24– 26]. The “no-go” theorem is introduced in Ref.[27] and elaborated further by, e.g. Ref.[28]. The phase transition of a model with a chemical potential is discussed by Eastham and Littlewood [29, 30]. As hinted to in the discussion on dynamic superradiance, the Dicke model is integrable, and this integrability extends even to the case of a distribution of two-level system energies:

X X g H = σz + ωa†a + a†σ− + aσ+ . (9.30) i i 2 i i i i

The integrability of this model can be proven by deﬁning a vector function L(z), having the following properties:

• The modulus of the vector L(z) is conserved.

• The Taylor expansion in z contains as many powers of z as there are two-level systems.

Together, these mean that L(z) contains suﬃcient conserved quantities that the system is integrable. This method is discussed in speciﬁc case of the Dicke model by Yuzbashyan, Kuznetsov, and Altshuler [31]. The history of this integrability, and its relation to a wider class of models is discussed in the review article by Dukelsky, Pittel, and Sierra [32]. The time-dependent solution, starting from the excited state is given by the same method used in question 9.2 by Barankov and Levitov [33].

Questions

Question 9.1: Asymptotic form of phase boundary

9.1.(a) By combining Eq. (9.16) with Eq. (9.17), show that at low densities, in the resonant case ( = ω), one may write:

−g T ' . (9.31) c ln (hMi/N)

9.1.(b) Find the low density asymptote in the more general case 6= ω.

Question 9.2: Non resonant Bloch vector precession Consider the non-resonant case of the Dicke model: g H = J z + a†J − + aJ + + ωa†a. (9.32) 2 9.5. FURTHER READING 87

9.2.(a) Show that the semiclassical equations of motion may be written as:

iJ˙− = J − − 2gαJ z, (9.33) iJ˙z = g J +α − J −α∗ , (9.34) iα˙ = gJ − + ωα. (9.35)

9.2.(b) By making a change to a rotating basis, but using α → αe−iηt, J − → J −e−iηt, show that this changes the above equations by replacing ω → ω˜ = ω − η and → ˜ = − η. The solution to these equations, analogous to the one discussed in Sec. 9.4 has the special property that α is real. Making the ansatz J − = z 2 Aα + iBα˙ , J = J0 − Cα , and using =(α) = 0, show the following:

9.2.(c) The imaginary part of the equation for J− implies A =B ˜ ;

9.2.(d) The equation for Jz implies C = gB/2;

9.2.(e) The equation for α implies gB/2 = 1, andω ˜ + ˜ = 0 The last of these conditions ﬁxes η, i.e. η = (ω + )/2. This leaves only the real part of the equation of motion for J−.

9.2.(f) Show that this equation has a ﬁrst integral, and that this ﬁrst integral is equivalent to the condition that |J|2 = J +J −+(J z)2 is a constant of the motion.

2 2 9.2.(g) Using the initial condition |J| = J0 , show that this ﬁnal equation may be written as:

g q ˜2 α˙ = α α2 − α2, α2 = 2J − 4 (9.36) 2 0 0 0 g

9.2.(h) Thus, writing α = α0 sin(θ/2), show that the equation of motion has the form:: " # g2 ˜2 θ¨ = J − 2 sin(θ) (9.37) 2 0 g

9.2.(i) Find the form of J z in terms of θ, and compare to the solution in the resonant case.

Lecture 10

Lasers and micromasers

In this lecture, we move to discuss a somewhat more classical system, the laser. The laser is classical in the sense that far above the lasing threshold, it produces large coherent states with classical correlations, and it is based on stimulated emission of radiation, which can be described classically. However, we will discuss not only simple lasers, for which reasonable classical descriptions may exist, but also more quantum mechanical systems, such as the micromaser, in which two-level atoms fall, one at a time, through a cavity, and in which the quantum mechanical evolution of each atom matters.

10.1 Density matrix equations for a micromaser and a laser

The aim of this section will be to write down a density matrix equation describing a laser, and to ﬁnd the steady state of this equation. The laser will consist of a cavity mode, coupled both to the continuum of modes outside the cavity, and to a gain medium that provides photons to the cavity mode. The modes outside the cavity lead to the standard decay term: d κ † † † ρ = − a aρ − 2aρa + ρa a , (10.1) dt decay 2 which can be written in the Fock basis as:

d κ h 0 p 0 i ρnn0 = − (n + n )ρnn0 − 2 (n + 1)(n + 1)ρn+1,n0+1 . (10.2) dt decay 2 For the gain medium, we will begin our discussion by discussing a formalism that can apply both to few atom lasing systems, and also to the typical incoherent gain medium, which will be introduced in the next section.

Gain from two-level atoms To give a model of gain that encompasses both coherent dynamics of a single injected atom (for a micromaser), as well as describing incoherent evolution of an externally pumped atom (for a laser), we will describe the

89 90 LECTURE 10. LASERS AND MICROMASERS

r t=0 lasing }transition γ γ κ r (fast) t=τ (fast) Figure 10.1: Left: Micromaser scheme, atoms in their excited state are injected at a rate r, and fall through the cavity in time τ. Right: Relation between a many-level lasing scheme, with decay via intermediate states to the ground state, and pumping to the excited state, and the quasi two-level description, with injection rate r, and two-level system lifetimes distributed according to the decoherence rate γ.

dynamics of the gain medium, allowing for arbitrary rates of decoherence and pumping. We therefore consider a two-level system, which starts in the excited state, and interacts with the cavity for time τ; new two-level systems are injected at a rate r. If we take τ to be the same for all atoms, we expect to see periodic dependence on τ, as one allows a given number of Rabi oscillations, transferring energy between the two-level system and the cavity. This model is appropriate for atoms falling through a cavity at a fixed speed. If instead τ is drawn from a Poisson distribution, average rate γ, then one recovers the standard laser result; at a rate γ the two-level system decays to some other states, and then rapidly decays to the ground state, and is then re-excited to the excited state of the two-level manifold at a rate r: the net result is loss of coherence at a rate γ, and injection of energy at a rate r. This relation is illustrated in Fig. 10.1. To find the gain supplied by the two-level systems, we want to find the evolution of the field density matrix. If interaction between the two- level system and the cavity for time τ causes the change ∆ρfield(τ), and new excited two-level systems are created at rate r, then one can write the evolution of the field for either a fixed τ, or a Poissonian distribution as

Z ∞ d d −γτ ρfield = r∆ρfield(τ), ρfield = r γdτ∆ρfield(τ)e . (10.3) dt τ dt γ 0

Thus, we require the change of the ﬁeld state due to evolution for time τ:

∆ρﬁeld(τ) = Tratom [ρ(t0 + τ)] − ρﬁeld(t0)

= ρfield,↓↓(t0 + τ) + ρfield,↑↑(t0 + τ) − ρfield,↑↑(t0). (10.4)

In the second line we have used the fact that the two-level system is by deﬁnition excited at t = t0. Now, using results of Sec. 3.2, we can follow 10.1. DENSITY MATRIX EQUATIONS FOR A MICROMASER AND A LASER 91 the evolution of a state |↑, ni with n photons to time τ:

Ω Ω |↑, ni → cos n+1 τ + i cos(2θ) sin n+1 τ |↑, ni 2 2 Ω + i sin(2θ) sin n+1 τ |↓, n − 1i , (10.5) 2 √ p 2 2 with Ωn = ( − ω) + g n, tan(2θn) = g n/( − ω) as in chapter 3. For simplicity, let us assume resonance, = ω, so θ = π/4. Then, using this wavefunction evolution, one ﬁnds:

√ √ h g g 0 i ∆ρ 0 (τ) = ρ 0 cos τ n + 1 cos τ n + 1 − 1 nn nn 2 2 √ √ g g 0 + ρ 0 sin τ n sin τ n . (10.6) n−1,n −1 2 2

It is apparent this equation describes gain, since it describes transfer of probability from the state with n − 1 photons to the state with n photons. In distinction Eq. (10.2) describes evolution from n + 1 photons to n photons. Note also that as anticipated, the ﬁxed time τ would lead to possible cancellation if integer numbers of Rabi oscillations occur. However, since the oscillation period is diﬀerent for each number of photons, the general result is very complicated, and can lead to a very non-smooth probability distribution.

Gain medium with decoherence — laser limit Now let us consider the case where τ is a random variable, distributed according to exponential decay, rate γ. Thus, we should average Eq. (10.6) over such a distribution. For simplicity, we will present here only the calculations for the diagonal elements, pn = ρnn; for the general case, see Q. 10.1.

√ dpn h D 2 g E D 2 g √ E i = r − sin τ n + 1 pn + sin τ n pn−1 , (10.7) dt gain 2 2 where:

D α E Z α γ Z sin2 τ = γdτe−γτ sin2 τ = − dτe−γτ eiατ + e−iατ − 2 2 2 4 γ 1 1 2 α2/2γ2 = − + − = . (10.8) 4 γ + iα γ − iα γ 1 + α2/γ2

Thus, the rate of gain due to a such a distribution of atomic coherence times is given by:

2 2 2 2 dpn r (g n/γ ) (g (n + 1)/γ ) = 2 2 pn−1 − 2 2 pn . (10.9) dt gain 2 1 + (g n/γ ) 1 + (g (n + 1)/γ ) 92 LECTURE 10. LASERS AND MICROMASERS

10.2 Laser rate equations

Combining the gain in Eq. (10.9) with the diagonal part of the decay in Eq. (10.2), one ﬁnds the equation of motion for the probability distribution:

dp r (g2n/γ2) (g2(n + 1)/γ2) n = p − p dt 2 1 + (g2n/γ2) n−1 1 + (g2(n + 1)/γ2) n

− κ [npn − (n + 1)pn+1] . (10.10) From this equation we can now determine when lasing will occur, as well as the full probability distribution when either well above or well below threshold.

Laser threshold condition

Lasing occurs when the vacuum state becomes unstable, i.e. when ∂thni > 0. Thus, let us ﬁnd: X ∂thni = n∂tpn. (10.11) n Let us however assume that g2n/γ2 is small — i.e. that the average population is small compared to (γ/g)2, so that the denominators in the gain can be expanded. Then,

X r g2n g2n g2(n + 1) g2(n + 1) ∂ hni = n 1 − p − 1 − p t 2 γ2 γ2 n−1 γ2 γ2 n n − κ [npn − (n + 1)pn+1] . (10.12)

Now, shifting the various sums, one can rewrite this as ( " X r g2 ∂ hni = p (n + 1)2 − n(n + 1) t n 2 γ2 n # ) g2 2 − (n + 1)3 − n(n + 1)2 − κ n2 − n(n − 1) , (10.13) γ2 which can be written as: r g2 r g2 r g2 2 ∂ hni = − κ hni + − h(n + 1)2i. (10.14) t 2 γ2 2 γ2 2 γ2 This can be understood as three terms; the first describes competition between loss and stimulated emission; the second describes spontaneous emission; the third describes the nonlinear susceptibility of the two-level systems, reducing the gain they provide when hni is large. The first and third of these could be recovered from a semiclassical theory of lasing, however the existence of spontaneous emission requires a fully quantum theory. However, if our aim is the lasing condition, this can easily be identified from the first term: stimulated emission outstrips decay if g2 > 2κ(γ2/r), and lasing then occurs. 10.3. LASER LINEWIDTH 93

Steady state probabilities The coefficients in Eq. (10.10) depend on n in such a way that one may write: dp n = −B p + B p − C p + C p , (10.15) dt n n n−1 n−1 n n n+1 n+1 2 2 2 with excitation rate Bn = rg (n+1)/[γ +g (n+1)] and decay rate Cn = κn. To find a steady state that is valid for all n, we need to find a relation between terms on the right hand side such that this will always vanish. The only consistent equality is to choose:

Bnpn = Cn+1pn+1 ⇐⇒ Bn−1pn−1 = Cnpn. (10.16) The steady state distribution is then given by:

n n 2 2 Y Bm−1 Y r(g /γ ) p = p = p , (10.17) n C 0 2κ [1 + (g2m/γ2)] 0 m=1 m m=1 with p0 set by normalisation. This product can in general be expressed in terms of a hypergeometric function. Let us consider two limits where a simpler expressions result.

Below threshold If we are at small powers, then small values of n will be most probable. In this case, one may neglect the term involving g2m/γ2 in the denominator. Thus, at low pumping rates, below threshold, one has 2 2 n pn ∝ (rg /2γ κ) , which is a thermal distribution, with exp(−βEeﬀ) = (rg2/2γ2κ), from which one recovers

1 r(g2/γ2) hni = = , (10.18) eβEeﬀ − 1 2κ − r(g2/γ2) in agreement with the small pumping rate limit of Eq. (10.14). Clearly, such an approximation is only valid below threshold, otherwise pn cannot be normalised. In this low pump limit, there is an “eﬀective pump rate”, limited by the ratio of g/γ, indicating that emission competes with relaxation.

Far above threshold The opposite limit occurs when hni is large enough that the term 1 in the denominator can be neglected. In this case one has n pn ∝ (r/2κ) /n!. This is a Poissonian distribution, as one would expect for a coherent state, with average population r/2κ, set only by the balance of pump rate and decay rate, since strong ﬁelds mean that stimulated emission beats any other relaxation process, thus for every excited atom created, half a photon is emitted.

10.3 Laser Linewidth

Our analysis of the laser so far was based on the density matrix equation and its steady states. Just as in lecture 5, we can also calculate the spectrum, by 94 LECTURE 10. LASERS AND MICROMASERS using two-time correlation functions via the quantum regression theorem. In the current context, we have a density matrix equation for the cavity ﬁeld, and so we can proceed directly to evaluate:

Z ∞ I(ν) ∝ 2< dteiνtha†(t)a(0)i (10.19) 0

As in the case of resonance ﬂuorescence, we can thus ﬁnd this by determining the expectation of ha†(t)i, when the initial state at t = 0 is taken to result from acting with a on the equilibrium density matrix. If we have the time-dependent solution of the equation of motion for the density matrix, then we can write √ † h † i X ha (t)i = Tr a ρ(t) = n + 1ρn,n+1(t). (10.20) n

k We will deﬁne the shorthand ρn,n+k(t) = ρn(t), and so our task is to ﬁnd 1 ρn(t). Before writing its equation of motion, let us anticipate how it should behave. Regardless of the initial conditions, at long times this expectation should decay to zero, as it does not conserve the number of photons. How- ever, since the initial conditions are those set by aρeqbm, the expectation of ha†(t)i will be non-zero; we are interested in how this function decays in time.

Equation of motion Combining the oﬀ-diagonal gain found in Eq. (10.31) with the decay term from Eq. (10.2) gives an equation of motion for the oﬀ-diagonal density matrix that can be written in the form:

dρk n = −Ak ρk + Bk ρk + Ck ρk . (10.21) dt n n n−1 n−1 n+1 n+1

Writing η = g2/γ2 for brevity, the coeﬃcients are given by: " # r ηp(n + 1)(n + 1 + k) Bk = , n 2 1 + η(n + 1 + k/2) + η2k2/16 k p Cn = κ n(n + k), r η(n + 1 + k/2) + η2k2/8 k Ak = + κ n + . n 2 1 + η(n + 1 + k/2) + η2k2/16 2

(see question 10.1 for derivation of the oﬀ-diagonal gain terms). Note that k k the terms Bn,Cn have been written in such a way that for k = 0 they reduce to the previous Bn,Cn discussed above. Whereas for k = 0, the k k k 0 0 0 term An is simply related to the Bn,Cn by An = Bn + Cn, no such simple relation exists in general. Since it was this simple relation that lead to the detailed balance condition for the probability distribution, no equivalent condition exists with k 6= 0. 10.3. LASER LINEWIDTH 95

Approximate solution Although the previous detailed balance solution does not apply to k 6= 0, it motivates a possible extension, which gives a solution decaying in time:

n k k Y Bm−1 ρk (t) = e−Dnt ρk, (10.22) n Ck 0 m=1 m

k where Dn is a parameter to be determined self consistently. Substituting k this ansatz for ρn into Eq. (10.21) gives:

" k # k k C k k k k k −(Dn−1−Dn)t n k − Dnρn = −Anρn + Bn−1 e k ρn Bn−1 " k # k k B k −(Dn+1−Dn)t n k + Cn+1 e k ρn (10.23) Cn+1

k k The terms in brackets are the expressions for ρn±1 in terms of ρn from k k Eq. (10.22). If Dn ' Dn±1, then the exponential terms may be dropped, giving: k k k k Dn = An − Cn − Bn. (10.24) k k Thus, if we solve Eq. (10.24), and find it varies slowly with n, then D ' Dn defines the decay rate of the off-diagonal density matrix. We therefore want to find D1 to solve for the linewidth. If we assume we are far above 1 1 threshold, so hni 1, then we can expand the square roots in Cn and Bn, to give:

r η(hni + 1 + 1/2) + η2/8 − η(hni + 1 + 1/2) D ' 2 ηhni + 1 + η(1 + 1/2) + η2/16 1 1 1 1 + κ hni + − hni 1 + − + O 2 2hni 8hni2 hni2 1 r g2 ' + κ (10.25) 8hni 2 γ2

Thus, using this time-dependence in the two-time correlation, we have:

−Dτ g1(τ) ' e ; (10.26)

We have been working in an interaction picture, neglecting free evolution of the photon ﬁelds at frequency ω. Restoring this, the frequency dependence of the intensity is then given by: D I(ν) = (10.27) (ν − ω)2 + D2

Hence the FWHM of this Lorentzian lineshape is given by:

1 r g2 2D = + κ (10.28) 4hni 2 γ2 96 LECTURE 10. LASERS AND MICROMASERS

Interpretation of linewidth This result for the lineshape has a simple interpretation in terms of phase diffusion. If we consider the radiation field of a laser to be described by an ensemble of coherent states, with effectively fixed magnitude |α| = phni, but different phases, then decay of correlations occurs because of growth of phase differences. Each time a photon is incoherently emitted emitted into, or lost from the cavity the phase changes. This is illustrated in Fig. 10.2. I(α )

θ R(α )

Figure 10.2: Cartoon of coherent state undergoing a random walk due to spontaneous emission events, leading to phase diﬀusion for the coherent state.

To turn this cartoon into a physical estimate, we start from the rate of such changes per unit time, which is d = κ + rg2/2γ2, by adding rates of photon decay and spontaneous emission (hence the appearance of the eﬀective pumping rate, as appearing in the threshold condition, rather than the total pump rate, as in the probability distribution far above threshold). If we assume a random walk, with d events per unit time, the change to the coherent state is governed by the random walk probability distribution: r td (∆α)2 P (∆α) = exp − (10.29) π td

However, the variation is entirely phase, so ∆α = phni∆θ. Using this, the decay of correlations after time τ can be found by averaging the state: ha†(t)a(0)i D E g (t) = = e−i[θ+∆θ(t)]eiθ 1 |hai|2 r Z td hni(∆θ)2 td = dθ exp −i(∆θ) − = exp − , (10.30) π td 4hni which reproduces the previous result, D = d/4hni.

10.4 Further reading

The laser rate equations presented here follow closely the presentation in Scully and Zubairy [10]. For a diﬀerent presentation of laser theory, the review article by Haken [34]. The discussion of laser linewidth in this Chapter is based on Scully and Zubairy [10]. 10.4. FURTHER READING 97

Questions

∗ Question 10.1: Oﬀ-diagonal matrix elements Evaluate the general oﬀ-diagonal matrix elements describing gain, de- k scribed in Eq. (10.6), and thus show that, writing ρn,n+k = ρn, one has:

k 2 2 2 4 dρn r (n + 1 + k/2)(g γ ) + k g /8 k = − 4 2 2 2 4 ρn dt gain 2 γ + (n + 1 + k/2)g γ + k g /16 " # r pn(n + k)(g2γ2) + ρk (10.31) 2 γ4 + (n + k/2)g2γ2 + k2g4/16 n−1

Question 10.2: Second-order correlation functions Using the quantum regression theorem, ﬁnd an expression for the second order correlation function: 1 D E g (τ) = a†(t)a†(t + τ)a(t + τ)a(t) (10.32) 2 N in terms of the equations of motion and initial conditions for the variables † pn = n|aρa |n . Solve these equations and initial conditions explicitly in the two limits g γ and g γ (i.e. far above, and far below threshold), and comment on the form of g2(τ) in the two cases. [For values near the threshold, one can solve these equations numerically to ﬁnd the full evolution of the second order coherence function.]

Lecture 11